U.S. patent number 4,650,424 [Application Number 06/754,920] was granted by the patent office on 1987-03-17 for educational device and method.
Invention is credited to Maurice E. Mitchell.
United States Patent |
4,650,424 |
Mitchell |
March 17, 1987 |
Educational device and method
Abstract
An educational toy and method for demonstrating characteristics
of a latticework of spacepoints including demonstrating (a) the
commonality of latticework between tetrahedron configuration
latticework and octahedron configuration latticework, (b) that
octahedron latticework merges with tetrahedron latticework, (c) the
13-plane structure of the common latticework, (d) how simultaneous
twinning in more than one of the 13 planes can form multitudes of
combinations of domains of tetrahedrons and octahedrons, and (e)
the altering of latticework by appropriately selecting the
dimensions of structure members that define spacepoints in the
latticework. Preferably, the structure members are similarly
dimensioned and oriented ellipsoidal elements which are gravity
stacked and optionally connectable and wherein the centerpoint of
each ellipsoidal element represents a spacepoint in the
latticework. With ellipsoidal elements, the latticework structure
is determined by the relative lengths of the three orthogonal axes
of symmetry of the ellipsoidal elements when the common axis and
the location of either orientation mark are known.
Inventors: |
Mitchell; Maurice E. (Walnut
Creek, CA) |
Family
ID: |
27028544 |
Appl.
No.: |
06/754,920 |
Filed: |
July 15, 1985 |
Related U.S. Patent Documents
|
|
|
|
|
|
|
Application
Number |
Filing Date |
Patent Number |
Issue Date |
|
|
628209 |
Jul 5, 1984 |
|
|
|
|
614050 |
May 25, 1984 |
|
|
|
|
430316 |
Sep 30, 1982 |
4461480 |
Jul 24, 1984 |
|
|
430315 |
Sep 30, 1982 |
|
|
|
|
Current U.S.
Class: |
434/211;
273/157R; 434/403; 446/85; 446/92; 52/DIG.10; 52/DIG.13 |
Current CPC
Class: |
A63F
9/12 (20130101); A63F 2009/1212 (20130101); Y10S
52/13 (20130101); Y10S 52/10 (20130101); A63F
2250/606 (20130101) |
Current International
Class: |
A63F
9/08 (20060101); A63F 9/06 (20060101); A63F
9/00 (20060101); G09B 023/04 () |
Field of
Search: |
;434/211,277,278,281,403
;273/157R ;446/85,92 ;52/DIG.10 |
References Cited
[Referenced By]
U.S. Patent Documents
Other References
Order in Space, by Keith Critchlow, The Viking Press, pp.
3-10..
|
Primary Examiner: Grieb; William H.
Attorney, Agent or Firm: Hall, Myers & Rose
Parent Case Text
RELATED APPLICATIONS
This is a continuation in part application of Ser. No. 430,315,
filed Sept. 30, 1982, now abandoned, of Ser. No. 430,316, filed
Sept. 30, 1982, now U.S. Pat. No. 4,461,480 granted July 24, 1984,
of Ser. No. 614,050, filed May 25, 1984, now abandoned, and of Ser.
No. 628,209, filed July 5, 1984, now abandoned.
Claims
I claim:
1. A method of teaching characteristics of latticework structure
comprising the steps of:
demonstrating the commonality of lattice structure of (a)
latticework arranged in accordance with a tetrahedron configuration
and (b) latticework arranged in accordance with a pyramid
configuration which has (i) a four-edge base and (ii) four faces
that extend from the base and meet at a point, said demonstrating
step including the steps of:
positioning a plurality of structural members relative to each
other to define spacepoints in a latticework arranged in accordance
with the tetrahedron configuration; and
positioning a plurality of structural members relative to each
other to define spacepoints in a latticework arranged in accordance
with the pyramid configuration;
wherein said positioning steps include:
merging together structural members along at least one face of the
latticework arranged in accordance with the tetrahedron
configuration with structural members along at least one
corresponding face of the latticework arranged in accordance with
the pyramid configuration to make the spacepoints along at least
one tetrahedron face coexistent with the spacepoints on the at
least one corresponding pyramid face.
2. A method according to claim 1 wherein each said positioning step
includes the step of:
gravity stacking a plurality of at least substantially similarly
dimensioned, similarly oriented ellipsoidal elements wherein each
ellipsoidal element is one of the structural members and the
centerpoint of each ellipsoidal element is a spacepoint in the
latticework;
all stacked ellipsoidal elements having at least substantially
similar dimensions.
3. A method according to claim 2 wherein said pyramid configuration
stacking step includes the step of:
stacking ellipsoidal elements to form a one-eighth octahedron
section that includes one face of a pyramid configuration; and
wherein said merging step includes the step of:
merging the pyramid face of the one-eighth octahedron section with
a face of the latticework arranged in accordance with the
tetrahedron configuration;
said merging resulting in at least a substantially uniform
latticework structure.
4. A method according to claim 3 wherein said demonstrating step
includes a further step of:
applying indicia to the ellipsoidal elements;
the indicia being applied and located on the stacked ellipsoidal
elements so that the indicia on the ellipsoidal elements display a
first identifiable pattern when the latticework is oriented to the
first bearing and a second identifiable pattern when the
latticework is oriented to a second bearing with a common axis
indicia that is oriented in the same bearing in said first bearing
and said second bearing.
5. A method according to claim 2 wherein said gravity stacking step
includes the step of stacking similarly dimensioned spheroids, the
centerpoint of each spheroid being a spacepoint in the
latticework.
6. A method according to claim 2 comprising the further step
of:
selecting the ellipsoidal element dimensions to have a major axis
and two minor axes of prescribed relative lengths that define the
latticework structure, the major axis and two minor axes
determining the distance between adjacent spacepoints in the
latticework.
7. A method as claimed in claim 6 wherein the positioning step
includes the step of:
forming the latticework in space of sufficient ellipsoidal elements
that the latticework spacepoints define thirteen nonparallel planes
in space, each plane being defined by a plurality of coplanar
ellipsoidal elements that contact a given ellipsoidal element.
8. A method according to claim 2 wherein said gravity stacking step
includes the stacking of magnetically interacting ellipsoids.
9. A method according to claim 2 wherein the positioning step
includes the step of:
forming the latticework in space of sufficient ellipsoidal elements
that the latticework spacepoints define thirteen nonparallel planes
in space, each plane being defined by a plurality of coplanar
ellipsoidal elements that contact a given ellipsoidal element.
10. A method according to claim 1 comprising a further step of:
selecting the dimensions of the structural members to determine
prescribed distances between each spacepoint in the latticework and
spacepoints adjacent to said given spacepoint, the dimensions of
the structural members which define inter-spacepoint distances
defining the latticework structure, said distances substantially as
set forth in Table II Sections (a) through (d), where numbered
spacepoints are those in FIG. 7.0 where the common axis is through
spacepoints 701 and 702 and the distance between spacepoints 701
and 702 is equal to unit distance `D`.
11. A method for teaching latticework characteristics comprising
the step of:
demonstrating the commonality of internal lattice structure between
similarly dimensioned, similarly oriented ellipsoidal elements
arranged to form (a) a tetrahedron configuration and (b) a pyramid
configuration having (i) a base and (ii) four sides, when such
configurations are extended in space;
wherein the commonality demonstrating step comprises the steps
of:
coupling ellipsoidal elements together to form a cuboctahedral type
configuration characterized by having twelve ellipsoidal elements
touching one ellipsoidal element;
orienting the ellipsoidal elements in said cuboctahedral type
configuration to a first prescribed bearing;
selectively stacking additional ellipsoidal elements relative to
the ellipsoidal elements that are coupled and oriented to the first
prescribed bearing to form the tetrahedron configuration;
orienting the ellipsoidal elements in said cuboctahedral type
configuration to a second prescribed bearing; and
selectively stacking additional ellipsoidal elements relative to
the ellipsoidal elements that are coupled and oriented to the
second prescribed bearing to form the pyramid configuration having
a base and four sides.
12. An educational device for teaching characteristics of
latticework structure comprising:
sets of structural members of suitable material, consisting of a
plurality of similarly dimensioned tetrahedral structural members
and a plurality of similarly dimensioned octahedral structural
members;
said structural members having suitable means for connecting
congruent faces to each other.
13. An educational device according to claim 12 wherein;
said structural members have suitable markings which indicate the
non-twinning orientation of each face of each pair of tetrahedral
structural members in relation to the appropriate congruent face of
their matching octahedral structural member.
14. An educational device according to claim 13 wherein;
the corner-to-corner distances of said structural members, being
essentially equal to the center-to-center distances of spacepoints
on said latticework structure where the corner-to-corner distances
are substantially equal to the ratios of unit distance `D` as set
forth in claim Table II Sections (a) through (d), where numbered
spacepoints are those in FIG. 7.0 where the common axis is through
spacepoints 701 and 702 and the distance between spacepoints 701
and 702 is substantially equal to unit distance `D`.
15. An educational toy for teaching characteristics of imaginary
thirteen nonparallel plane latticework structure comprising:
a plurality of similarly dimensioned ellipsoidal elements, each
ellipsoidal element being dimensionally characterized by three
orthogonal axes of symmetry and a curved surface in which every
plane cross section is an ellipse or a circle, one of said axes
being marked as a common axis with a suitable similar indicia on
the ellipsoidal surface indicating the correct orientation or
bearing of said common axis; and
each ellipsoidal element having a suitable similar indicia on the
ellipsoidal surface indicating the correct triangular or
tetrahedral orientation of the ellipsoidal element when correctly
gravity stacked on a gravity tray, or placed in an imaginary
thirteen nonparallel plane latticework structure; and
each ellipsoidal element having a suitable similar indicia on the
ellipsoidal surface indicating the correct pyramidal or octahedral
orientation of the ellipsoidal element when correctly gravity
stacked on a gravity tray, or placed in a imaginary thirteen
nonparallel plane latticework structure; and
each ellipsoidal element having six uniquely oriented polarized
connecting holes through the centerpoint thereof;
where said six uniquely oriented polarized connecting holes may
optionally be connected to an identical uniquely oriented polarized
connecting hole in a correctly oriented adjacent ellipsoidal
element, without either ellipsoidal element being removed from its
correct gravity stacked position on the gravity tray, with a
special torsion spring friction coupling device with the aid of a
special torsion spring friction coupling insertion tool;
said six uniquely oriented polarized connecting holes allowing a
similarly oriented corresponding ellipsoidal element to optionally
be connected to each polarized end, thus twelve similarly oriented
corresponding ellipsoidal elements may optionally be connected to
one central corresponding ellipsoidal element in the shape of a
simple cuboctahedral type configuration, using special torsion
spring friction couplers and a special torsion spring friction
insertion tool.
16. An educational toy according to claim 15 wherein ellipsoidal
elements of said plurality are connected to form a latticework
having a tetrahedral configuration with the cuboctahedral type
configuration as a nucleus portion thereof; and
wherein ellipsoidal elements of said plurality are connected to
form a latticework having a five-sided pyramid configuration with
the cuboctahedral type configuration as a nucleus portion
thereof.
17. An educational toy according to claim 16 further
comprising:
a tray for supporting said ellipsoidal elements connected in either
configuration;
a side of said either configuration lying on said tray when
supported thereby.
18. An educational toy according to claim 17 wherein said tray
includes a surface and a walled corner on the surface, said walled
corner being characterized by a front vertical wall and a vertical
side wall that is perpendicular to said front vertical wall, said
surface being inclined toward said walled corner.
19. An educational device according to claim 18 wherein the said
tray includes a said vertical side wall that may be positioned at
an angle to said front vertical wall.
20. An educational toy according to claim 18 wherein each
ellipsoidal element has indicia thereon which orient the
ellipsoidal element with relation to the said perpendicular side
wall of said walled corner of said tray regardless of which
configuration the said ellipsoidal element is in and a second and
third indicia thereon which face one direction when said
ellipsoidal elements are supported on said tray in one
configuration and which face another direction when said
ellipsoidal elements are supported on said tray in the other
configuration.
21. An educational toy according to claim 20 wherein the indicia on
each ellipsoidal element includes a triangle, a square and a circle
positioned on each ellipsoidal element; and
wherein said triangle on each ellipsoidal element lies parallel to
said tray surface when said ellipsoidal elements are supported in a
tetrahedral configuration; and
wherein said square on each ellipsoidal element lies parallel to
said tray surface when said ellipsoidal elements are supported in a
pyramid configuration; and
wherein center of said circle points the same away in relation to
said perpendicular side wall when said ellipsoidal elements are
supported in either the said tetrahedral configuration or the said
pyramid configuration.
22. An educational device according to claim 20 wherein the
centerpoints of said ellipsoidal elements define spacepoints;
and
wherein said ellipsoidal elements are connected by suitable means
while being supported on the said tray so that the spacepoints are
characterized by a geometric latticework structure formed of only
octahedron sections and tetrahedron sections merged together.
23. A device according to claim 22 wherein a number of spacepoints
form a rhombohedron geometric latticework which includes two
tetrahedrons and an octahedron positioned therebetween, a face of
each tetrahedron lying coextensively against a corresponding face
of said octahedron.
24. A device according to claim 23 wherein;
one edge of one tetrahedron with spacepoints numbered 701 and 702
in FIG. 7.0 has been designated the common axis edge,
said edge having a length from corner-to-corner substantially equal
to unit distance `D` and the other edge lengths from
corner-to-corner substantially equal to the ratio of said unit
distance `D` as set forth in Table II Sections (a) through (d),
where numbered spacepoints are those in FIG. 7.0.
25. An educational device according to claim 20 wherein said
ellipsoidal elements are positioned to form a rhombohedral group
with each edge of said rhombohedral group comprising an equal
number of said elements, said equal number being at least three,
and
from said rhombohedral group forming an "up" tetrahedral group, a
"down" tetrahedral group and an octahedral group similar to FIG.
7.0, and
demonstrating that the "up" tetrahedral group merges correctly with
the octahedral group in only four unique ways, and
demonstrating that the "down" tetrahedral group merges correctly
with the octahedral group in only four unique ways, and
demonstrating that none of the four general faces of the "up"
tetrahedral group can be correctly merged with any of the four
general faces of the "down" tetrahedral group.
26. An educational device according to claim 25 comprising two said
rhombohedral groups, and
from said rhombohedral groups forming two "up" tetrahedral groups,
two "down" tetrahedral groups and two octahedral groups, and
demonstrating that none of the four general faces of one "up"
tetrahedral group can be merged correctly with any of the four
general faces of the other "up" tetrahedral group, or with any of
the four general faces of the "down" tetrahedral groups, and
demonstrating that none of the four general faces of one "down"
tetrahedral group can be merged correctly with any of the four
general faces of the other "down" tetrahedral group, or with any of
the four general faces of the "up" tetrahedral groups, and
demonstrating that none of the eight general faces of one
octahedral group be can be merged correctly with any of the eight
general faces of the other octahedral group, thus
demonstrating that twinning of the general latticework structure
occurs when a general face of one tetrahedral group is merged with
a similar general face of an essentially similar second tetrahedral
group, and
demonstrating that twinning of the latticework structure occurs
when a general face of one octahedral group is merged with a
similar general face of an essentially similar second octahedral
group.
27. A device according to claim 15 wherein said three orthogonal
axes of symmetry are of such lengths that the center-to-center
distances of the ellipsoidal elements are substantially equal to
the ratios as set forth in Table I Sections (a) through (d), where
the ellipsoids in Table I are as shown in FIG. 4.2.
28. An educational device according to claim 27 wherein said three
orthogonal axes of symmetry are of such lengths that the
center-to-center distances of the ellipsoidal elements as shown in
FIG. 4.2, for the "Rhombus 30" ellipsoid set are as follows:
401 to 402 and 403 to 421 are dimensions substantially equal to the
unit distance `D`, and
401 to 421 and 403 are dimensions substantially equal to 1.61803
times the unit distance `D`, and
401 to 403 and 402 to 421 are dimensions substantially equal to
1.47337 times the unit distance `D`, and
are of such lengths that the center-to-center distances of the
ellipsoidal elements as shown in FIG. 4.2, for the "Isosceles 60"
ellipsoid set are as follows:
401 to 402 is a dimension substantially equal to the unit distance
`D`, and
401 to 403 and 403 to 402 are dimensions substantially equal to
0.89800 times the unit distances `D`, and
401 to 421, 402 to 421 and 403 to 421 are dimensions substantially
equal to 1.40126 times the unit distance `D`, and
are of such lengths that the center-to-center distances of the
ellipsoidal elements as shown in FIG. 4.2, for the "Edge 60"
ellipsoid set are as follows:
401 to 402 is a dimension substantially equal to the unit distance
`D`, and
403 to 421 is a dimension substantially equal to 1.53884 times the
unit distance `D`, and
401 to 421 and 421 to 402 are dimensions substantially equal to
1.40126 times the unit distance `D`, and
401 to 403 and 403 to 402 are dimensions substantially equal to
0.95106 times the unit distance `D`.
29. An educational toy according to claim 15 wherein the set
comprises thirteen spheres arranged in a cubotahedron configuration
of spheres.
30. An educational toy according to claim 15 further
comprising:
a first plurality of twenty-two separate spheres stackable with the
cuboctahedron configuration to form a regular tetrahedron
configuration.
31. An educational toy according to claim 30 further
comprising:
a second plurality of seventeen separate spheres stackable with the
cuboctahedron configuration to form a regular pyramid configuration
with congruent sides.
32. An educational device for teaching chracteristics of imaginary
thirteen nonparallel plane latticework structure comprising:
a plurality of similarly dimensioned matching sets of corresponding
tetrahedral and octahedral structural elements, each said matching
set consisting of an `up` tetrahedral element, a matching `down`
tetrahedral element and a matching octahedral element;
each said matching set being dimensionally characterized by one
edge of the base of the `up` tetrahedral being marked as the common
axis with a suitable similar indicia indicating the correct
orientation or bearing of said edge of said matching set, said edge
having a length from corner-to-corner designated a unit distance
`D` and the other edge lengths from corner-to-corner designated as
a ratio of said unit distance `D` substantially as set forth in
Table II Sections (a) through (d), where numbered spacepoints are
those in FIG. 7.0;
each face of said matching set being marked with suitable similar
indicia indicating the correct orientation of that face in relation
to its corresponding face on the appropriate opposite matching
structural element of said set, in an imaginary thirteen
nonparallel plane latticework structure; and
each face of said matching set being fit with suitable means to
attach said face to either an appropriate congruent corresponding
face on a matching opposite structural element of said set or of a
similar set, when no twining is occurring in the latticework
structure; and
each face of said matching set being fit with suitable means to
attach said face optionally to a congruent face on a similar
matching structural element of said set or of a similar set when
twinning of the common latticework is being demonstrated.
33. An educational device according to claim 32 wherein the edge
distances between the numbered corner spacepoints in FIG. 7.0 for
the "Rhombus 30" tetrahedron and octahedron set are as follows:
701 to 702, 703 to 721, 704 to 722, 723 to 724, 803 to 804, 804 to
822, 822 to 821 and 821 to 803 are dimensions substantially equal
to each other, and
701 to 721, 702 to 703, 704 to 724, 722 to 723, 802 to 803, 803 to
823, 823 to 822 and 822 to 802 are dimensions substantially equal
to each other, and
701 to 703, 702 to 721, 704 to 723, 722 to 724, 802 to 804, 804 to
823, 823 to 821 and 821 to 802 are dimensions substantially equal
to each other, and
where thirty corner spacepoints numbered 802 in FIG. 7.0 can be
merged together in one point making solid around said one point,
and
the edge distances between the numbered corner spacepoints in FIG.
7.0 for the "Isosceles 60" tetrahedron and octahedron set are as
follows:
701 to 702, 723 to 724, 803 to 804 and 821 to 822 are dimensions
substantially equal to each other, and
701 to 703, 703 to 702, 723 to 722, 722 to 724, 803 to 802, 802 to
804, 821 to 823 and 823 to 822 are dimensions substantially equal
to each other, and
701 to 721, 702 to 721, 703 to 721, 722 to 704, 723 to 704, 724 to
704, 802 to 821, 821 to 803, 803 to 823, 823 to 804, 804 to 822 and
822 to 802 are dimensions substantially equal to each other,
and
where sixty corner spacepoints numbered 721 can be merged together
in one point making a solid around said one point, and
the edge distances between the numbered corner spacepoints in FIG.
7.0 for the "Edge 60" tetrahedron and octahedron set are as
follows:
701 to 702, 723 to 724, 803 to 804 and 821 to 822 are dimensions
substantially equal to each other, and
703 to 721, 704 to 722, 803 to 821 and 804 to 822 are dimensions
substantially equal to each other, and
701 to 721, 721 to 702, 723 to 704, 704 to 724, 803 to 823, 823 to
804, 821 to 802 and 803 to 822 are dimensions substantially equal
to each other, and
701 to 703, 703 to 702, 723 to 722, 722 to 724, 803 to 802, 802 to
804, 821 to 823 and 823 to 822 are dimensions substantially equal
to each other, and
where sixty corner spacepoints numbered 721 can be merged together
in the point making a solid around said one point.
34. An educational device according to claim 32 wherein the edge
distances between the numbered corner spacepoints in FIG. 7.0 for
the "Rectangular Rotation" tetrahedron and octahedron sets are as
follows:
701 to 702, 703 to 721, 704 to 722, 723 to 724, 803 to 804, 803 to
821, 804 to 822 and 821 to 822 are dimensions substantially equal
to each other, and
703 to 701, 703 to 702, 721 to 701, 721 to 702, 722 to 723, 722 to
724, 704 to 723, 704 to 724, 823 to 803, 823 to 804, 823 to 821,
823 to 822, 802 to 803, 802 to 804, 802 to 821 and 802 to 822 are
dimensions substantially equal to each other, and
where a given whole number of three or nore corner spacepoints
numbered 802 in FIG. 7.0 can be merged together with two times the
said given wholenumber of corner spacepoints numbered 701 in FIG.
7.0 in one point making a solid around said one point, and
the edge distances between the numbered corner spacepoints in FIG.
7.0 for the "Triangular Rotation" tetrahedron and octahedron sets
are as follows:
701 to 702, 702 to 703, 703 to 701, 723 to 724, 724 to 722, 722 to
723, 821 to 822, 822 to 823, 823 to 821, 802 to 803, 803 to 804 and
804 to 802 are dimensions substantially equal to each other,
and
721 to 701, 721 to 702, 721 to 703, 704 to 722, 704 to 723, 704 and
724, 802 to 821, 802 to 822, 803 to 831, 803 to 823, 804 to 822 and
804 to 823 are dimensions substantially equal to each other,
and
where a whole number of corner spacepoints numbered 721 can be
merged together in one point making a solid around said one point,
and
the edge distances between the numbered corner spacepoints in FIG.
7.0 for the "Rectangular Ellipsoid" tetrahedron and octahedron sets
are as follows:
701 to 702, 723 to 724, 821 to 822 and 803 to 804 are dimensions
substantially equal to each other, and
721 to 703, 704 to 722, 803 to 821, and 804 to 822 are dimensions
substantially equal to each other, and
721 to 701, 721 to 702, 703 to 701, 703 to 702, 722 to 723, 722 to
724, 704 to 723, 704 to 724, 802 to 821, 802 to 822, 802 to 803,
802 to 804, 823 to 821, 823 to 822, 823 to 803 and 823 to 804 are
dimensions substantially equal to each other, and
where a first given whole number of three or more corner
spacepoints numbered 823 in FIG. 7.0 can be merged together with
two times the said first given whole number of corner spacepoints
numbered 721 in FIG. 7.0 in one point making a solid around said
one point, and
where a different second given whole number of four or more corner
spacepoints numbered 823 in FIG. 7.0 can be merged together with
two times the said second given whole number of corner spacepoints
numbered 701 to FIG. 7.0 in a second point making a solid around
said second point.
35. An educational device according to claim 32 wherein the edge
distances between the numbered corner spacepoints in FIG. 7.0 for
the "General Ellipsoid" tetrahedron and octahedron sets are as
follows:
701 to 702, 723 to 724, 803 to 804 and 821 to 822 are dimensions
substantially equal to each other, and
701 to 703, 722 to 724, 821 to 823 and 802 to 804 are dimensions
substantially equal to each other, and
702 to 703, 722 to 723, 802 to 803 and 822 to 823 are dimensions
substantially equal to each other, and
701 to 721, 704 to 724, 802 to 822 and 803 to 823 are dimensions
substantially equal to each other, and
702 to 721, 704 to 723, 802 to 821 and 804 to 823 are dimensions
substantially equal to each other, and
703 to 721, 704 to 722, 803 to 821 and 804 to 822 are dimensions
substantially equal to each other, and
where corners of four tetrahedrons and corners of three octahedrons
may be merged together at one point making a plane surface passing
through said one point, said corners being numbered 701, 702, 703,
704, 802, 803 and 804 in FIG. 7.0.
36. An educational method comprising the step of:
demonstrating the commonality of internal lattice structure between
equal diameter spheroids arranged to form (a) a regular tetrahedron
configuration and (b) a pyramid configuration having (i) an
equilateral base and (ii) four congruent sides, when such
configurations are extended in space;
wherein the commonality demonstrating step includes the steps
of:
forming the tetrahedron configuration of spheroids and the pyramid
configuration of spheroids to have the same number of layers;
and
coupling at least one side of the pyramid configuration to a
corresponding one of the tetrahedron faces comprising the step of
defining the spheroids along each said at least one side of the
pyramid configuration to be the spheroids along each corresponding
tetrahedron face.
37. An educational method according to claim 36 wherein commonality
demonstrating step comprises the further step of:
dividing the pyramid into four equal 1/8th octahedron sections with
two planes passing diagonally across and perpendicular to the
pyramid base, spheroids common to a plurality of the 1/8th
octahedron sections being represented in each such 1/8th octahedron
section as whole spheroids.
38. An educational method according to claim 37 wherein the
coupling comprises the further step of:
merging each pyramid side of each 1/8th octahedron section to a
corresponding tetrahedron face.
39. An educational device for teaching characteristics of
latticework structure comprising:
sets of structural members of suitable material, comprising a
plurality of similarly dimensioned tetrahedral structural members
and a plurality of similarly dimensioned octahedral structural
members, where the ratio of said structural members in said sets is
essentially two tetrahedral structural members for each octahedral
structural member; and
said sets of structural members having a designated common axis
edge on one edge of the tetrahedrons, which edge has a length
substantially equal to the unit distance `D`; and
said sets of structural members having corner-to-corner dimensions
substantially as set forth in Table II Sections (a) through (d);
and
an additional plurality of corresponding structural members
comprising one-half octahedron structural members made by passing a
single plane through any one of the three planes with four
spacepoints therein, resulting in a structural member containing
five spacepoints of the original six spacepoints of the
corresponding octahedron structural member; and
an additional plurality of corresponding structural members
comprising first one-quarter octahedron structural members made by
passing two planes through any two of the three planes with four
spacepoints therein, resulting in a structural member containing
four spacepoints of the original six spacepoints of the
corresponding octahedron structural member; and
an additional plurality of corresponding structural members
comprising second one-quarter octahedron structural members made by
passing two planes through any two opposite spacepoints, with each
plane passing through a different third spacepoints equidistance on
the edges made by the other four spacepoints of the original six
spacepoints of the corresponding octahedron structural member,
resulting in a structural member containing three spacepoints of
the original six spacepoints of the corresponding octahedron
structural member; and
an additional plurality of corresponding structural members
comprising first one-eighth octahedron structural members made by
passing three planes through the three planes with four spacepoints
therein, resulting in a structural member containing three
spacepoints of the original six spacepoints of the corresponding
octahedron structural member; and
an additional plurality of corresponding structural members
comprising one-half tetrahedron structural members made by passing
a plane through two of the four spacepoints of the tetrahedron and
a third spacepoint equidistance on the edge defined by the two
remaining spacepoints of the said corresponding tetrahedron
structural member, resulting in a structural member containing
three spacepoints of the original four spacepoints of the
corresponding tetrahedron structural member; and
said structural members having suitable means for connecting
congruent faces to each other; and
said structural members having suitable markings which indicate the
proper orientation of each face of each structural member in
relation to the face of each other structural member.
40. An educational device according to claim 39 demonstrating the
shape of the effective ellipsoid of influence in imaginary thirteen
nonparallel plane space latticework structure when said ellipsoid
of influence is expanded into the interstices therebetween
wherein;
an additional plurality of corresponding structural members
comprising said second one-eighth octahedron structural members
made by passing four planes that are parallel to the original four
sets of parallel planes of the original octahedron and equidistance
between those corresponding sets of parallel planes, resulting in
six structural members each containing one spacepoint of the
original six spacepoints of the corresponding octahedron structural
member; and
an additional plurality of corresponding structural members
comprising one-quarter tetrahedron structural members made by
passing six planes into the center of a corresponding tetrahedron
structural member where each plane passes through a point on each
edge equidistance from first two spacepoints on ends of said edge
and tangent to the corresponding ellipsoids touching at said point,
each said plane stopping when it intersects another said plane,
resulting in four structural members each containing one spacepoint
of the original four spacepoints of the corresponding tetrahedron
structural member; and
said structural members having suitable means for connecting
congurent faces to each other; and
said structural members having suitable markings which indicate the
proper orientation of each face of each structural member in
relation to the face of each other structural member.
41. An educational device according to claim 40 wherein:
an additional plurality of corresponding structural members
comprising merged combinations of two or more tetrahedral and
octahedral structural members where no twinning has occurred.
42. An educational device according to claim 41 wherein:
an additional plurality of corresponding structural members
comprising merged combinations of two or more tetrahedral and
octahedral structural members where twinning is occurring.
43. An educational method comprising the steps of:
demonstrating the commonality of internal lattice structure between
equal diameter spheres arranged to form (a) a regular tetrahedron
configuration and (b) a pyramid configuration having (i) an
equilateral base and (ii) four congurent sides, when such
configurations are extended in space;
wherein the commonality demonstrating step comprises the further
steps of:
packing equal diameter spheres relative to each other to form a
cuboctahedron configuration;
orienting the spheres in the cubotahedron configuration in a first
prescribed manner;
selectively stacking additional equal diameter spheres relative the
spheres that are packed and oriented in said first prescribed
manner to form a regular tetrahedron configuration;
orienting the spheres in the cuboctahedron configuration in a
second prescribed manner; and
selectively stacking additional equal diameter spheres relative to
the spheres that are packed and oriented in said second prescribed
manner to form a pyramid configuration having a four-sided
equilateral base and four equal sides.
44. A method of teaching characteristics of latticework structure
comprising the steps of:
demonstrating the commonality of lattice structure of (a)
latticework extending from a basic tetrahedron first configuration
and (b) latticework extending from a basic pyramid second
configuration which has a (i) four-edged base and (ii) four sides
that extend from the base and meet at a point, said demonstrating
step including the steps of:
positioning a plurality of structure members relative to each other
to define spacepoints in a latticework arranged in one of the two
basic configurations;
adding structural members to expand the latticework arranged in
said one basic configuration; and
removing structural members from the expanded latticework to define
spacepoints in a latticework arranged in the other of the two basic
configurations.
45. A method according to claim 44 wherein said positioning step
includes the step of:
gravity stacking a plurality of at least substantially similarly
dimensioned, similarly oriented ellipsoidal elements, wherein each
ellipsoidal element is one of the structural members and the
centerpoint of each ellipsoidal element is a spacepoint in the
latticework.
46. An educational device for teaching characteristics of
latticework structure comprising:
sets of structural members of suitable material, comprising a
plurality of similarly dimensioned tetrahedral structural members
and a plurality of similarly dimensioned octahedral structural
members, said structural members having congruent faces and means
for connecting congruent faces of said structural members
together.
47. An educational device according to claim 46 wherein said
sturctural members are positioned to form a helix.
48. An educational device according to claim 46 wherein said
structural members are positioned to form a spiral.
49. An educational device according to claim 46 wherein said
structural members are positioned to form a helical like
spiral.
50. An educational device according to claim 46 wherein said
structural members are positioned to form a tetrahedron.
51. An educational device according to claim 46 wherein said
structural members are positioned to form an octahedron.
52. An educational device according to claim 46 wherein said
structural members are positioned to form a rhombohedron.
53. An educational device according to claim 46 wherein said
structural members are positioned to form a geometrical
configuration.
54. An educational device according to claim 46 wherein said
structural members are positioned to form a three dimensional
geometric form.
55. An educational device for teaching characteristics of imaginary
thirteen nonparallel plane latticework structure comprising:
a plurality of ellipsoidal elements of substantially equal size and
shape with similar dimensions and indicia, and
each said element having a common axis essentially passing through
its center with a suitable common axis indicia at one end, and
each said element having a suitable rotation indicia placed
perpendicular to said common axis on the surface of said element,
and
demonstrating that when said elements are positioned with a similar
bearing with one end of each common axis touching the oppostie end
of the common axis of an adjacent said element, and
where all common axes are parallel and all rotational indicia have
the same bearing, when twinning does not occur, the centers of said
elements define one unique latticework structure, wherein
twinning does not occur when elements are oriented with the same
bearing and positioned so their common axes are touching, and
first three equal sets of elements positioned along their common
axes are merged together so that each element is touching three
other elements, and
each additional element is positioned touching at least four other
elements where said four other elements are in a plane, or
each additional element is positioned touching at least one other
common axis, or
each additional element is positioned touching at least three other
elements where said three other elements are touching each other in
a plane, and no other element is already touching said three other
elements.
56. An educational device according to claim 55 wherein the
elements have a complex ellipsoidal shape where all planes through
the center of said complex ellipsoidal element cut the surface of
element in the form of segments of a circle or segments of an
ellipse merged together so that a line tangent to the surface of
the element is also tangent to both segments where the segments
meet.
Description
BACKGROUND OF THE INVENTION
In the past, ellipsoids with equal axes have been closely packed,
under the force of gravity, into various structures. Critchlow in
his book Order in Space (1970) illustrates ellipsoids with equal
axes arranged in (a) a simple 4-ellipsoid with equal axes
tetrahedral configuration, (b) a simple 6-ellipsoid with equal axes
octahedral configuration, and (c) a simple 13-ellipsoid with equal
axes cuboctahedral configuration, referring to each simple
configuration as a distinct regular pattern. In discussing these
arrangements, Critchlow indicates that the tetrahedral
configuration is the most economic grouping of--ellipsoids of equal
axes--while "the next most economic regular grouping of--ellipsoids
of equal axes--is six in the octahedral configuration."
A preliminary examination of the ellipsoids of equal axes arranged
in a tetrahedral configuration with a triangular base and in a
4-sided pyramid configuration with a square base would appear to
support Critchlow's characterizations and distinctions. The lines
connecting the centerpoints of the three ellipsoids of equal axes
which form the "base" of the simple 4-ellipsoid tetrahedron form an
equilateral triangle. On the other hand, the lines connecting the
centerpoints of the four ellipsoids of equal axes which form the
"base" of a simple 5-ellipsoid pyramid (i.e. a one-half octahedron)
form a square.
In the past, a lattice structure based on a tetrahedral
configuration and a lattice structure based on a pyramidal, or
one-half octahedral configuration were viewed as different.
The fact that a tetrahedral configuration, an octahedral (or
pyramidal) configuration and a cuboctahedral configuration yield
precisely the same lattice structure when extended into space or
merged together, however, has remained unknown and conspicuously
unsuggested, especially as applied to ellipsoids of influence under
the influence of gravity.
SUMMARY OF THE INVENTION
It is an object of the invention to demonstrate to a student that
given a plurality of ellipsoids of influence closely packed under
the following four conditions;
(a) ellipsoids of essentially equal size and shape;
(b) oriented with a similar bearing;
(c) stacked under the influence of gravity;
(d) with at least one common axis; then:
(I) the latticework structure started with four ellipsoids in a
simple tetrahedral configuration; is equal to
(II) the latticework structure started with five ellipsoids in a
simple pyramidal or one-half octahedral configuration; is equal
to
(III) the latticework structure started with thirteen ellipsoids in
a simple cuboctahedral configuration;
when these simple latticework structures of ellipsoids of influence
are extended into space.
That is, notwithstanding the fact that the base of the simple
tetrahedron latticework structure has a triangular base, the base
of the simple octahedron latticework structure has a rectangular
base (one-half octahedron), and the simple cuboctahedron
latticework structure could be said to have both triangular bases
and rectangular bases, the present invention demonstrates that,
over space, ellipsoids of influence, arranged by starting in either
of the three simple patterns, given the four conditions, closely
pack in the same way.
Further, it is an object of the invention to show that a large
tetrahedral configuration formed of, for example, ellipsoids,
comprises the same internal latticework structure as a large
pyramidal (one-half octahedron) configuration formed of the same
ellipsoids, and that both of these configurations comprises the
same internal latticework structure as a large cuboctahedral
configuration formed of the same ellipsoids.
It is yet another object of the invention to demonstrate that in
(a) a tetrahedral configuration having a base, or face, of fifteen
ellipsoids (e.g. five ellipsoids along each edge) and (b) a
pyramidal configuration having a base of twenty-five ellipsoids in
a 5.times.5 arrangement, the same 13-ellipsoid cuboctahedral type
of configuration is embodied in each. Moreover, in that the
cuboctahedral type of configuration of closely packed ellipsoids is
common to both the tetrahedral and octahedral configurations, a
student will recognize the commonality of latticework structure of
the three `heretofore different` latticeworks, when these
latticework structures are extended into space under the influence
of gravity.
It is thus a further object of the invention to demonstrate the
commonality of the closely packed tetrahedral, octahedral and
cuboctahedral configurations of ellipsoids by selectively
assembling or disassembling (a) a tetrahedral configuration of
ellipsoids and (b) an octahedral configuration of ellipsoids with
an intact cuboctahedral type of configuration of closely packed
ellipsoids contained therein.
Furthermore, it is an object of the invention to show that the
tetrahedral configuration closely packed and expanded into space
under the aforementioned four conditions define imaginary thirteen
nonparallel planes.
Still further, where a latticework is defined by spacepoints that
are determined by the centerpoints of ellipsoids or other
corresponding structural members representing fields of influence
that are closely packed under the aforementioned four conditions,
then the relative dimensions of the major and minor axes of the
ellipsoid when the common axis and the location of either
orientation mark are known, uniquely deterine the relative
distances or lengths between the spacepoints in the corresponding
latticework structure.
Conversely, the relative distances or lengths between the four
corners of a corresponding tetrahedron, when the edge that is equal
to the common axis is known, uniquely determine the major and minor
axes and the location of both orientation marks of the
corresponding ellipsoid or ellipsoidal field of influence that
creates the corresponding latticework structure and uniquely
determine the relative distances or lengths between, and
orientation of, the six corners of the corresponding octahedron and
uniquely determine the relative distances or lengths between, and
orientation of, the four corners of the other inverted
corresponding tetrahedron.
Further still, when the common axis and the location of either
orientation mark are known, the lengths of the major and minor axes
of the corresponding ellipsoid or ellipsoidal field of influence,
uniquely define imaginary thirteen nonparallel planes in the
corresponding latticework structure when the corresponding
ellipsoids are gravity stacked under the aforementioned four
conditions.
A further object of the invention is to demonstrate that eight
ellipsoids closely packed under the aforementioned four conditions
uniquely define two corresponding tetrahedrons plus their
corresponding octahedron;
this invention thus demonstrates that two corresponding
tetrahedrons plus their corresponding octahedron equal one
corresponding rhombohedron;
and it is further shown that each corresponding rhombohedron is
equal in volume to six corresponding tetrahedrons;
further still, it is shown that each corresponding octahedron is
equal in volume to four corresponding tetrahedrons;
it further demonstrates that the total solid angles of the eight
corners of two corresponding tetrahedrons plus the solid angles of
the six corners of their corresponding octahedron equal the total
solid angle in the centerpoint of one corresponding ellipsoid;
still further, it is shown that there is one general common
imaginary thirteen nonparallel plane space latticework that closely
packed ellipsoids of influence assume when the aforementioned four
conditions are satisfied, where the ellipsoid of influence can be
imagined to be in a plurality of ellipsoids in the form of;
(A) a crystal or solid;
(B) a liquid;
(C) a gas; or
(D) very regular spirals helical like of ellipsoids in a radio wave
or some other electromagnetic spectrum wave.
Methods that achieve these objects are exemplified by the following
methods.
A method teaching the characteristics of corresponding latticework
structure comprises the steps of demonstrating the commonality of
lattice structure of (a) latticework arranged in accordance with a
tetrahedral configuration and (b) latticework arranged in
accordance with a pyramidal configuration (one-half octahedron)
which has (i) a four-edge base and (ii) four faces that extend from
the base and meet at a point, the demonstrating step including the
steps of: positioning a plurality of structural members relative to
each other to define spacepoints in a latticework arranged in
accordance with the tetrahedral configuration; positioning a
plurality of structural members relative to each other to define
spacepoints in a latticework arranged in accordance with the
pyramidal configuration; wherein the positioning steps include
merging together structural members along at least one face of the
latticework arranged in accordance with the tetrahedral
configuration with structural members along at least one
corresponding face of the latticework arranged in accordance with
the pyramidal configuration to make the spacepoints along at least
one tetrahedral face coexistent with the spacepoints on at least
one corresponding pyramidal face. Each positioning step includes
the step of gravity stacking a plurality of at least substantially
similarly dimensioned similarly oriented ellipsoidal elements,
wherein each ellipsoidal element is one of the structural members
and the centerpoint of each ellipsoidal element is a spacepoint in
the latticework.
The educational toy of the invention is exemplified by a toy for
teaching characteristics of latticework structure comprising; a
plurality of similarly dimensioned ellipsoidal elements, each
ellipsoidal element being dimensionally characterized by a major
axis and two minor axes where the axes are orthogonal and are axes
of symmetry;
and each ellipsoidal element being characterized by having one end
of the common axis marked with a circle or other indicia to
indicate the orientation of the common axis;
and each ellipsoidal element being characterized by a common axis
connector hole passing through the centerpoint of this common axis
orientation mark and the centerpoint of the ellipsoidal element and
through the ellipsoidal element and thus uniquely defining the
common axis of the ellipsoidal element;
and each ellipsoidal element being characterized by a triangular
orientation mark or indicia on the surface of the ellipsoidal
element locating the up direction when a first ellipsoidal element
is gravity stacked on the gravity tray starting in the tetrahedral
configuration;
and each ellipsoidal element being characterized by optionally
being capable of being connected to a second ellipsoidal element
along their common axis connector holes when both ellipsoidal
elements have been gravity stacked on the gravity tray and oriented
so that their orientation marks all point the same way with the
common axis orientation mark pointing away from the lowest corner
of the gravity tray, without being moved from their gravity stacked
positions by using a special connecting tool and special torsion
spring friction coupling;
and each ellipsoidal element being characterized by a second
connector hole such that when a third ellipsoidal element is
gravity stacked against first and second ellipsoidal elements that
have optionally been connected along their common axis connector
holes, and all orientation marks on the three ellipsoidal elements
are correctly oriented in the same direction, and said third
ellipsoidal element is gravity stacked so that it touches the first
ellipsoidal element at one point and the second ellipsoidal element
at one point and the gravity tray at one point, the centerline of
the aforementioned second connector hole passes through the
centerpoint of the third ellipsoidal element and the centerpoint of
the first ellipsoidal element, thus enabling the third ellipsoidal
element to optionally be connected to the first ellipsoidal element
using their second connector holes and the aforementioned special
torsion spring friction coupler;
and each ellipsoidal element being characterized by a third
connector hole such that when a third ellipsoidal element is
optionally connected to a first ellipsoidal element along their
second connector holes, and the first ellipsoidal element is
optionally connected to a second ellipsoidal element along their
common axis connector holes on the gravity tray as aforementioned,
the centerline of the third connector holes passes through the
centerpoints of the third ellipsoidal element and the second
ellipsoidal element in such a manner that the third ellipsoidal
element optionally may be connected to the second ellipsoidal
element using their third connector holes and aforementioned
torsion spring friction device;
and each ellipsoidal element being characterized by a fourth
connector hole such that when a fourth ellipsoidal element is
gravity stacked on top of optionally connected first three
ellipsoidal elements, with its common axis orientation mark in the
same direction as the common axis orientation marks of the first
three ellipsoidal elements, with all triangular orientation marks
in the up position, the centerline of the fourth connector hole
passes through the centerpoints of the fourth ellipsoidal element
and the third ellipsoidal element in such a manner that the fourth
ellipsoidal element optionally may be connected to the third
ellipsoidal element using their fourth connector holes and
aforementioned torsion spring friction device;
and each ellipsoidal element being characterized by a fifth
connector hole such that when a fourth ellipsoidal element is
gravity stacked on top of optionally connected first three
ellipsoidal elements, with its common axis orientation mark in the
same direction as the common axis orientation marks of the first
three ellipsoidal elements, with all triangular orientation marks
in the up position, the centerline of the fifth connector hole
passes through the centerpoints of the fourth ellipsoidal element
and the first ellipsoidal element in such a manner that the fourth
ellipsoidal element optionally may be connected to the first
ellipsoidal element using their fifth connector holes and
aforementioned torsion spring friction device;
and each ellipsoidal element being characterized by a sixth
connector hole such that when a fourth ellipsoidal element is
gravity stacked on top of optionally connected first three
ellipsoidal elements, with its common axis orientation mark in the
same direction as common axis orientation marks of the first three
ellipsoidal elements, with all triangular orientation marks in the
up position, the centerline of the sixth connector hole passes
through the centerpoints of the fourth ellipsoidal element and the
second ellipsoidal element in such a manner that the fourth
ellipsoidal element optionally may be connected to the second
ellipsoidal element using their sixth connector holes and
aforementioned torsion spring friction device;
and each ellipsoidal element being characterized by a rectangular
orientation mark or square mark or indicia indicating the up
position of the ellipsoidal element when gravity stacked in the
pyramidal or one-half octahedral configuration by optionally
connecting the common axis connecting holes of the first and second
ellipsoidal elements after being gravity stacked on the gravity
tray, and further optionally connecting the common axis connecting
holes of the third and fourth ellipsoidal elements that have also
been gravity stacked on the gravity tray, and further optionally
connecting the fourth connecting holes of the first and third
ellipsoidal elements after rotating their triangular orientation
marks toward the front wall of the gravity tray so that their
rectangular orientation marks are in the up position, and further
optionally connecting the fourth connecting holes of the second and
fourth ellipsoidal elements after rotating their triangular
orientation marks toward the front wall of the gravity tray so that
their rectangular orientation marks are in the up position;
and each ellipsoidal element being characterized by being able to
be optionally connected to the first four ellipsoidal elements that
have been correctly gravity stacked on the gravity tray in the
rectangular or pyramidal configuration, by being considered the
fifth ellipsoidal element and being correctly oriented and gravity
stacked on top of the first four ellipsoidal elements, so that the
second connecting holes of the fifth and first ellipsoidal elements
are in alignment, the third connecting holes of the fifth and
second ellipsoidal elements are in alignment, the fifth connecting
holes of the fifth and third ellipsoidal elements are in alignment
and the sixth connecting holes of the fifth and fourth ellipsoidal
elements are in alignment, thus allowing optional connection of the
fifth ellipsoidal element to any or all of the first four
ellipsoidal elements without moving any of the ellipsoidal elements
from their gravity stacked positions on the gravity tray.
Thus the plurality of ellipsoids have six unique connector holes
similarly located through their centerpoints and have similarly
located common axis orientation marks, similarly located triangular
or tetrahedron orientation up marks and similarly located
rectangular orientation or square pyramidal up marks, where the six
connector holes are made in such a manner that the ellipsoidal
elements optionally may be connected without disturbing the gravity
stacked position of the ellipsoidal elements on the gravity tray
when the ellipsoidal elements are gravity stacked properly in
either the tetrahedral configuration or the pyramidal (one-half
octahedron) configuration.
Alternately the unique connector hole endpoints may be velcro or
magnetic elements that have a polarity effect to them as it is seen
that the unique connector holes are effectively polarized. For
example, it is necessary to have opposite endpoints of the same
connector hole touching each other before even the same unique
connector holes can be correctly optionally connected. Another
method to connect the gravity stacked ellipsoidal elements uses
multiple suction cups made of soft rubber or other flexible
material on each side of an effective zero length coupling device
that optionally can be placed at the contact points as the elements
are being gravity stacked.
Table I gives four different types of ellipsoids sets in Sections
(a) through (d). It is logical to dimension the ellipsoid sets and
their corresponding tetrahedron and octahedron block sets using a
method that give similar ratios for similar distances if such a
method exists. One such method is to define the center-to-center
distance between the first four ellipsoids that can be gravity
stacked on the tray in the simple tetrahedron configuration. In
FIG. 4.2, the said four ellipsoids are 401, 8402, 403 and 421. Then
use these same center-to-center distances as the corner-to-corner
distances between the spacepoints of the corresponding tetrahedrons
and octahedrons as shown in FIG. 7.0. This causes the distances
between spacepoints 701 and 702 to be the same as the distances
between centerpoints 401 and 402; between spacepoints 702 and 703
to be the same as between centerpoints 402 and 403; between
spacepoints 703 and 701 to be the same as between centerpoints 403
and 401; between spacepoints 721 and 701 to be the same as between
centerpoints 421 and 401; between spacepoints 721 and 702 to be the
same as between centerpoints 421 and 402; and between spacepoints
721 and 703 to be the same as between centerpoints 421 and 403.
This method of dimensioning enables the exact same ratios of
distances to be used in Table I Sections (a) through (d) for
center-to-center distances for ellipsoids 401, 402, 403 and 421,
and in Table II Sections (a) through (d) for corner-to-corner
distances for spacepoints 701, 702, 703 and 721 for the
corresponding matching `up` tetrahedron and the corresponding
spacepoints for the matching `down` tetrahedron and the
corresponding spacepoints for the matching octahedron as shown in
FIG. 7.0.
It is not necessarily intuitively clear that these six distances
uniquely define the space latticework of a plurality of ellipsoids
of influence closely packed under the aforementioned four
conditions and this is part of the subject matter of the present
invention.
In Table I Sections (a), (b), (c) and (d), the common axis of the
ellipsoidal element is defined by the centerpoints of ellipsoids
401 and 402 as indicated by the common axis circle marks to the
righthand end of the said ellipsoids. The length of the first axis
of the ellipsoid is equal to the distance between the centerpoints
of ellipsoids 401 and 402. For example, one-half of the first axis
is from centerpoint of 401 to the point of contact with 402, plus
the other one-half of the first axis is from said point of contact
to the centerpoint of 402.
In all four Sections of Tables I and II, for a given set of
ellipsoids or their matching corresponding tetrahedron and
octahedron set of blocks, the distance between the centerpoints of
ellipsoids on the common axis has arbitrarily been assigned the
unit distance `D`, so that the other center-to-center distances and
other corner-to-corner spacepoint distances may be defined as a
ratio of the common axis length unit distance `D`.
In Table I Section (a) all six center-to-center distances are equal
to the unit distance `D`. The length of the first axis of the
ellipsoid, as stated above, is equal to the center-to-center
distance of ellipsoids 401 and 402. The student sees by examining
FIG. 5.2 that ellipsoid 421 of FIG. 4.2 closely packs against 401,
402 and 403 in such a manner that the center line between 421 and
403 is always parallel to the gravity tray in the rectangular
orientation and is always at right angles to the common axis 401
and 402. Thus, in Table I Section (a), all of the ellipsoids have
to be ellipsoids of rotation about a third axis that is vertical to
the gravity tray in the rectangular configuration. The vertical
view of the ellipsoid set in Table I Section (a) in the rectangular
configuration are circles and look just like FIG. 5.2. The length
of the second axis is equal to the length of the first axis and is
equal to the distance between ellipsoids 421 and 403 in the
ellipsoid set in Table I Section (a).
In Table I Section (a), the third axis of the ellipsoid has to be
perpendicular to the plane of the first and second axes, but may be
any length desired. This enables the student to calculate the
length of the third axis to obtain the desired center-to-center
distance of the four remaining equal center-to-center distances
which desired distance is set forth in Table I Section (a) as a
ratio of the unit distance `D`. In this case, the remaining four
center-to-center distances are equal to the unit distance `D`.
This, of course, is the special case where the ellipsoid is a
sphere.
In Table I Section (b) the center-to-center distance of ellipsoids
401 and 402 is equal to unit distance `D` and as aforementioned
this is the length of the first axis of the ellipsoid., The student
sees by examining FIG. 5.2 that ellipsoid 421 of FIG. 4.2 closely
packs against 401, 402 and 403 in such a manner that the center
line between the centerpoints of 421 and 403 is always parallel to
the gravity tray in the rectangular orientation and is always at
right angles to the first common axis 401 and 402. Therefore the
length of the second axis is equal to the distance between the
centerpoints of 421 and 403 and from Table I Section (b) is also
equal to the unit distance `D`.
Thus, in Table I Section (b), all of the sets of ellipsoids have to
be ellipsoids of rotation about the third axis that is vertical to
the gravity tray in the rectangular configuration. Therefore the
vertical view of the ellipsoid sets in Table I Section (b) in the
rectangular configuration are circles and look just like FIG.
5.2.
Therefore for the ellipsoid sets in Table I Section (b), the third
axis of the ellipsoid is perpendicular to the plane of the first
and second axes, just as the definition of an ellipsoid requires,
but this third axis may be any length desired and still be an
ellipsoid of revolution. This enables the student to adjust the
length of the third axis of each set of ellipsoids to obtain the
desired center-to-center distance of the remaining four equal
center-to-center distances as set forth as a ratio of the unit
distance `D`, in Table I Section (b).
In Table I Section (c) the center-to-center distance of ellipsoids
401 and 402 is equal to unit distance `D` and as aforementioned
this is the length of the first axis of the ellipsoid. Also, the
center-to-center distances of ellipsoids 402 and 403, and 403 and
401 are all equal to unit distance `D`. Therefore, in the
triangular configuration, the centerpoints of these three
ellipsoids make an equilateral triangle. The ellipsoid sets that
make this center-to-center pattern have to be sets of ellipsoids of
revolution where the third axis is the axis of rotation and is
perpendicular to the gravity tray in the triangular or tetrahedron
configuration.
In Table I Section (c), the length of the second axis of the
ellipsoid is equal to the distance between ellipsoids 401 and 402,
as these sets of ellipsoids are ellipsoids of revolution about the
third axis perpendicular to the gravity tray in the tetrahedron
configuration. Therefore the second axis is equal to the unit
distance `D`, and is at right angles to the common axis and in the
plane of the gravity tray when the ellipsoidal element is in the
tetrahedron configuration. The vertical view of these sets of
ellipsoids then look just like FIG. 4.2 as their cross-section are
circles in the tetrahedron configuration.
Therefore for the ellipsoid sets in Table I Section (c), the third
axis of the ellipsoid is perpendicular to the plane of the first
and second axes, just as the definition of an ellipsoid requires,
but this third axis may be any length desired and still be an
ellipsoid of revolution. This enables the student to adjust the
length of the third axis of each set of ellipsoids to obtain the
desired center-to-center distance of the remaining three equal
center-to-center distances as set forth as a ratio of the unit
distance `D`, in Table I Section (c).
In Table I Section (d) the center-to-center distance of the common
axis ellipsoids 401 and 402 is equal to the length of the first
axis of the ellipsoid, and is also equal to unit distance `D`. The
student sees by examining FIG. 5.2 that ellipsoid 421 of FIG. 4.2
closely packs against 401, 402 and 403 in such a manner that the
center line between 421 and 403 is always parallel to the gravity
tray in the rectangular orientation and is always at right angles
to the common first axis 401 and 402. Therefore the length of the
second axis is equal to center-to-center distance of ellipsoids 421
and 403.
However, in Table I Section (d), the center-to-center distance
between ellipsoids 421 and 403 is not equal to the center-to-center
distance between ellipsoids 401 and 402. Thus, in Table I Section
(d), we have unique sets of general ellipsoids that are not
ellipsoids of rotation.
Thus, the sets of ellipsoids in Table I Section (d), have first and
second axes that are at right angles to each other in the
rectangular configuration. Therefore, the third axis of these sets
of ellipsoids has to be vertical to the plane of the gravity tray
when the ellipsoidal element is in the rectangular configuration,
and can be any length desired. As set forth in Table I Section (d),
the length of the first axis of each set of ellipsoids is equal to
the center-to-center distance between ellipsoids 401 and 402 and is
equal to unit distance `D`, and the length of the second axis is
equal to the center-to-center distance between ellipsoids 421 and
403 and given as a ratio of unit distance `D`. The length of the
first axis and the length of the second axis uniquely define the
ellipse made by passing a plane through the centerpoint of the
ellipsoid that is also parallel to the gravity tray in the
rectangular or octahedral configuration. This enables the student
to determine the length of one axis of a vertical ellipse made by
passing a plane through the centerpoints of ellipsoids 401, 403 and
601 in FIG. 5.2. The second axis of this vertical ellipse is also
equal to the third axis of the ellipsoid. This enables the student
to adjust the length of the third axis of each set of ellipsoids to
obtain the desired center-to-center distance of the remaining four
equal center-to-center distances as set forth as a ratio of the
unit distance `D`, in Table I Section (d).
Also the toy is exemplified by a toy wherein the corner-to-corner
edge distances of corresponding tetrahedron blocks and a
corresponding octahedron on block are equal to the center-to-center
distances of adjoining gravity stacked corresponding ellipsoidal
elements in a corresponding space latticework. The corners of said
corresponding tetrahedron blocks and said corresponding octahedron
block are selected to form a corresponding latticework structure.
There is a simple friction or torsion spring or velcro or pressure
sensitive or suction cup or magnetic coupling arrangement in the
four faces of each pair of corresponding tetrahedron blocks and in
the eight faces of each corresponding octahedron block. This simple
coupling arrangement enables the face in question to be connected
to an exact corresponding face with equal edge lengths of either
another corresponding tetrahedron block or another corresponding
octahedron block.
Further, it is an object of the invention to demonstrate that
latticework twinning in one plane may occur with any corresponding
latticework, using corresponding tetrahedron blocks and
corresponding octahedron blocks.
Still further, it is an object of the invention to demonstrate
simultaneous latticework twinning in several of the imaginary
thirteen nonparallel planes at the same time. All of the unique
ratios of center-to-center distances and their corresponding
corner-to-corner distances of corresponding tetrahedron blocks and
corresponding octahedron blocks, set forth in Table I and Table II,
demonstrate latticework structures that allow simultaneous twinning
in at least two of the imaginary thirteen nonparallel planes of the
basic latticework structure.
The corresponding tetrahedron and octahedron block sets optionally
have orientation marks on their faces so that the `up`
corresponding tetrahedron and each of its four faces can be
uniquely distinguished from its corresponding matching `down`
tetrahedron and each of its four faces and so that their
corresponding matching octahedron may be uniquely oriented in
relation to said pair of corresponding tetrahedrons and each of
said octahedron's eight faces can be uniquely identified and
oriented in relation to the eight matching faces on said pair of
corresponding tetrahedron blocks. Further, indicia showing the
polarity of spacepoints on said tetrahedron blocks and octahedron
blocks optionally may be added.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1.0 is an illustration of a geometric ellipsoid.
FIG. 1.1 is an illustration of an ellipsoid generated by rotating
an ellipse about a minor axis resulting in an oblate ellipsoid.
FIG. 1.2 is an illustration of an ellipsoid generated by rotating
an ellipse about the major axis resulting in a prolate
ellipsoid.
FIG. 1.3 is an illustration of an ellipsoid generated by rotating a
circle about its diameter making an ellipsoid with equal axes.
FIG. 2.0 is a top view of an ellipsoidal element in the triangular
or tetrahedral configuration with the triangular mark 111 in the up
direction when the gravity tray is in the plane of the sheet of
paper with the circle mark 109 to the right.
FIG. 2.1 is the right hand view of the ellipsoidal element in FIG.
2.0.
FIG. 2.2 is a top view of the said ellipsoidal element as in FIG.
2.0 but with the triangular mark rotated toward the viewer so that
the rectangular mark 113 is in the up direction when the gravity
tray is in the plane of the sheet of paper.
FIG. 2.3 is the right hand view of said ellipsoidal element in FIG.
2.2.
FIG. 3.0 is an isometric view of the gravity tray 301 onto which
ellipsoidal elements may be stacked. The tray 301 includes a
surface 303 which is inclined toward a corner 305. Two walls 311
and 313 disposed atop surface 303 meet at the corner 305 to define
a walled corner.
FIG. 3.1 is an isometric view of the special torsion spring
friction coupler 321, together with the special torsion spring
inserting device 351 with a special torsion spring removing hook
353.
FIG. 3.2 is a top view of the gravity tray 301 with the first three
ellipsoidal elements in a triangular or tetrahedral configuration
shown in cross-section in that plane that passes through the common
axis connector hole and the second connector hole and the third
connector hole, with the special torsion spring inserting device
351 in the process of optionally inserting a special torsion spring
friction coupler 321 between the third ellipsoidal element and the
first ellipsoidal element along the second connector hole. Special
torsion spring friction couplers 321 have already been used to
optionally connect the common axis connector holes of the first
ellipsoidal element and the second ellipsoidal element and to
optionally connect the third connector holes of the third
ellipsoidal element and the second ellipsoidal element.
FIG. 4.0 is a top view of the gravity tray with five ellipsoidal
elements with their common axis connecting holes in alignment as
indicated by the circle marks 109 pointing to the right away from
the lowest corner 305 of the gravity tray 301 and touching the
lefthand wall 313 at 202 and touching the front wall 311.
FIG. 4.1 is a top view of the gravity tray with nine ellipsoidal
elements in a triangular configuration as indicated by the
triangular marks 111 being in the up position. This view shows the
center line of the common axis connector holes between the
centerpoints of ellipsoidal elements 401 and 402; the center line
of the second connector holes between the centerpoints of
ellipsoidal elements 403 and 401; and the center line of the third
connector holes between the centerpoints of ellipsoidal elements
403 and 402.
FIG. 4.2 is a top view of the gravity tray with thirteen
ellipsoidal elements in a triangular tetrahedral configuration that
enables the student to see that the fourth connector holes pass
through the centerpoints of ellipsoidal elements 421 and 403; and
to see that the fifth connector holes pass through the centerpoints
of ellipsoidal elements 421 and 401 and to see that the sixth
connector holes pass through the centerpoints of the ellipsoidal
elements 421 and 402.
FIG. 4.3 is a top view of FIG. 4.2 with three more ellipsoidal
elements added that enables the student to see that ellipsoidal
element 410 is very similar to ellipsoidal element 402 but moved
over two rows of ellipsoidal elements.
FIG. 4.4 is a top view of FIG. 4.3 with three more ellipsoidal
elements added that enables the student to see that ellipsoidal
elements 410 and 423 are gravity stacking in the same vertical
plane as ellipsoidal element 402.
FIG. 4.5 is a top view of FIG. 4.4 with three more ellipsoidal
elements added that enables the student to see that ellipsoidal
elements 410, 411, 412, 423, 424, 902, 906, 907 and 908 are all in
the same plane in a 3.times.3 configuration of the rectangular or
square or pyramid (one-half octahedron) configuration. This enables
the student to see the plane that the rectangular marks 113 should
be in as indicated.
FIG. 4.6 is a top view of FIG. 4.5 with eight more ellipsoidal
elements added.
FIG. 4.7 is a top view of FIG. 4.6 with the last five ellipsoidal
elements added to complete the tetrahedral configuration with faces
with five ellipsoidal elements on each edge.
FIG. 4.8 is a view of the completed tetrahedron in FIG. 4.7 that
has been rotated forward about the common axis represented by
ellipsoidal elements 401, 402, 405, 406 and 407 in such a manner
that the tetrahedron edge represented by ellipsoidal elements 915,
914, 911, 905 and 415 is facing the viewer and is in the up
rectangular position as indicated by the rectangular marks.
FIG. 4.9 is a view of FIG. 4.8 where ellipsoidal elements equal to
one-eighth of an octahedron with faces with edges equal to five
ellipsoidal elements have been added to show how a cube with
diagonals equal to five ellipsoidal elements is formed from the
tetrahedron in FIG. 4.8.
FIG. 5.0 is a top view of the gravity tray with five ellipsoidal
elements in the rectangular or pyramid or (one-half octahedron)
configuration as indicated by the rectangular marks being in the up
position.
FIG. 5.1 is a top view of FIG. 5.0 with twenty additional
ellipsoidal elements being added. The cross-section A--A is equal
to one-eighth of an octahedral configuration base where the base
has edges equal to five ellipsoidal elements or one-quarter of a
pyramid base where the base has edges equal to five ellipsoidal
elements, in a one-half octahedral configuration. This
cross-section A--A enables the student to see where the ellipsoidal
elements to change the tetrahedron of FIG. 4.8 to the cube of FIG.
4.9 are located in the one-eighth octahedron with face edges equal
to five ellipsoidal elements.
FIG. 5.2 is a top view of FIG. 5.1 with four more ellipsoidal
elements being added.
FIG. 5.3 is a top view of FIG. 5.2 with twelve more ellipsoidal
elements being added.
FIG. 5.4 is a top view of FIG. 5.3 with nine more ellipsoidal
elements being added to complete the third rectangular layer.
FIG. 5.5 is a top view of FIG. 5.4 with four more ellipsoidal
elements being added to complete the fourth rectangular layer.
FIG. 5.6 is a top view of FIG. 5.5 with one more ellipsoidal
element being added to complete the rectangular or pyramid
configuration with faces with edges equal to five ellipsoidal
elements, (one-half octahedron) configuration.
FIG. 6.0 is an isometric view of the cube with diagonals equal in
length to five ellipsoidal elements. This view enables the student
to better see how the seven ellipsoidal elements of the one-eighth
octahedron as indicated by cross-section A--A closely pack on the
tetrahedron of FIG. 4.7 and FIG. 4.8 to form the cube with
diagonals equal to five ellipsoidal elements in FIG. 4.9 and FIG.
6.0.
FIG. 6.1 is an isometric view of the cuboctahedron with ellipsoidal
elements designated with the letters A through M.
FIG. 7.0 is a top view of a corresponding `up` tetrahedron and its
matching inverted corresponding `down` tetrahedron and their
matching corresponding octahedron that connects said pair of
tetrahedrons. This view identifies the corner spacepoints that are
used to define the unique distance ratios between corner-to-corner
spacepoints as set forth in the four Sections of Table II for these
sets of matching corresponding tetrahedron and octahedron blocks.
Each unique set of blocks have multiple twinning planes in their
common imaginary thirteen nonparallel plane space latticework.
DESCRIPTION OF THE INVENTION
In FIG. 1.0, an ellipsoidal element 101 is shown. As is well-known
in geometry, the ellipsoidal element 101 has a major axis (along
the line y, in FIG. 1.0) and two minor axes (along the x line and
the z line, in FIG. 1.0). The ellipsoid can also be described as
having three orthogonal axes of symmetry.
In FIG. 2.0 an ellipsoidal element 200 (like ellipsoidal element
105 of FIG. 1.2 on its side) is shown as adapted for use in
accordance with the invention.
It is first noted that six unique connector holes pass through the
centerpoint of the ellipsoidal element 200. The common axis
connector hole center line is denoted by endpoints 201 and 202, and
is indicated by the common axis orientation circle mark 109. The
triangular orientation mark or tetrahedral configuration mark 111
indicates the up position of the ellipsoidal element 200 when
element 200 is in the triangular or tetrahedral configuration when
the plane of the paper is equal to the surface 303 of the gravity
tray as shown in FIG. 3.0.
The second connector hole center line is denoted by endpoints 203
and 204.
The third connector hole center line is denoted by endpoints 205
and 206.
The fourth connector hole center line is denoted by endpoints 207
and 208.
The fifth connector hole center line is denoted by endpoints 209
and 210.
The sixth connector hole center line is denoted by endpoints 211
and 212.
FIG. 2.1 is the righthand end view of FIG. 2.0.
FIG. 2.2 is the top view of the ellipsoidal element 200 when
element 200 is in the rectangular or pyramidal (one-half
octahedron) configuration as is indicated by the rectangular
orientation mark 113 being in the up position when the plane of the
paper is equal to surface 303 of the gravity tray in FIG. 3.0.
FIG. 2.3 is the righthand end view of FIG. 2.2.
Each of the six unique connector holes are located in such a manner
that only identical connector holes optionally may be correctly
coupled together. For example, the common axis connector holes
between two properly oriented adjacent ellipsoidal elements 200,
may optionally be coupled with the special torsion spring friction
coupler without moving the gravity stacked position of the two
elements 200 when properly oriented on the gravity tray 301 of FIG.
3.0 with the help of the special torsion spring friction coupler
inserting device 351 of FIG. 3.1. The second connector holes of two
properly oriented adjacent ellipsoidal elements 200 may optionally
be coupled with the special torsion spring friction coupler without
moving the gravity stacked position of the two elements 200 when
properly oriented on the gravity tray 301.
Conversely, it is not possible to correctly orient and couple the
common axis connector hole of one ellipsoidal element 200 with the
second connector hole of an adjacent ellipsoidal element 200 either
on or off of the gravity tray 301.
Alternately ellipsoidal elements 200 may be made without connector
holes, with the three orientation marks or indicia. In this
embodiment of the invention, magnetic elements, or velcro elements,
or pressure sensitive adhesive couplers, or multiple suction cup
couplers or other suitable coupling devices may optionally be used
to couple adjacent ellipsoidal elements 200.
The six unique connector holes are also polarized so that only
opposite ends of each unique connector hole can be correctly
oriented to be optionally connected. This enables those skilled in
the art to use a wide diversity of unique couplers in this
invention, and it is an object of this invention to include all of
these suitable unique coupling techniques within the scope of this
invention.
Referring now to FIG. 3.0, a tray 301 is shown onto which
ellipsoidal elements may be stacked. The tray 301 includes a
surface 303 which is inclined toward a corner 305. Two walls 311
and 313 are disposed atop surface 303 and meet at the corner 305 to
define a walled corner.
Alternately, in other embodiments of the invention, wall 313 may be
positioned at different angles when gravity stacking complex
ellipsoidal elements of influence, where multiple combinations of
ellipsoidal surfaces are merged together and further where the
common axis connector hole is not aligned with any of the axes of
the ellipsoidal element.
In FIG. 4.7, a plurality of ellipsoidal elements embodied as
ellipsoids of equal axes are stacked to form a basic tetrahedral
configuration 400 which has four sides, or faces. Because
ellipsoids of equal axes are being gravity stacked, the basic
tetrahedral configuration is a regular tetrahedron of congruent
sides. Moreover, the ellipsoids of equal axes are stacked closely
packed where the aforementioned four conditions are satisfied as
their common axis orientation circle marks are all pointing away
from the lefthand wall 313, and their triangle orientation marks
111 or indicia are in the up direction from the surface 303 of tray
301. Due to the incline of the surface 303, the first ellipsoid 401
rests against the walled corner 305. The ellipsoids, it is noted,
are gravity stacked. That is, a plurality of ellipsoids --such as
those labelled 401, 402, 405, 406, 407, 409, 408, 404, 403, 410,
411, 412, 413, 414 and 415 form a bottom layer which rests on the
surface 303 (see FIGS. 4.3, 4.6 and 4.7). An ellipsoid is properly
oriented with its common axis orientation circle mark to the right
and its triangular orientation mark up is gravity stacked atop each
interstitial pocket between three ellipsoids in the lower layer to
form a next layer. In this way the ellipsoids are closely packed
under the aforementioned four conditions. In FIG. 4.7, ellipsoids
such as 421, 422, 900, 901, 902, 424, 423, 903, 904 and 905 and the
plane defined thereby form a second layer laying atop the lower
layer. Additional layers are similarly formed by further stacking.
The top ellipsoid 915, and the ellipsoids 401, 407 and 415 together
form the four corners of a basic tetrahedral configuration gravity
stacked five layers high. The six edges of the tetrahedral
configuration 400 include the following ellipsoids respectively:
(1) 401, 402, 405, 406 and 407; (2) 407, 409, 412, 414 and 415; (3)
401, 403, 410, 413 and 415; (4) 401, 421, 906, 912 and 915; (5)
407, 901, 908, 913 and 915; and (6) 915, 914, 911, 905 and 415.
Referring next to FIG. 4.8, the tetrahedral configuration 400 of
FIG. 4.7 is again shown, however, oriented with a different
bearing. Specifically, the tetrahedral configuration 400 is
oriented with the rectangular orientation marks or square indicia
facing upward. The thirty-five ellipsoids in tetrahedral
configuration 400 in FIG. 4.7 have been optionally coupled along
their six connector holes and rotated about the common axis of
ellipsoids 401, 402, 405, 406 and 407 in such a manner that the top
ellipsoid 915 is moved towards the bottom of FIG. 4.8. The numeral
labels on the ellipsoids in FIG. 4.7 and 4.8 --which may also be
provided on the ellipsoids in implementing the invention --aid in
correctly orienting and positioning the ellipsoids in each
bearing.
In FIG. 4.9, ellipsoids are added to the tetrahedral configuration
400 oriented as in FIG. 4.8. The ellipsoids labelled US combine
with ellipsoids 905, 911, and 914 in order to form a four-edged
face with 3.times.3 ellipsoid edges, each ellipsoid on the face
having its square orientation indicia facing outward from the plane
of the paper and its common axis orientation circle mark or indicia
pointing to the right. Considering the 905-911-914 face as the base
of a 3.times.3 base pyramid, it is noted that the ellipsoids 903,
904, 910 and 909 form a next layer. An additional ellipsoid in the
interstitial pocket between ellipsoids 903, 904, 910 and 909
completes a 3.times.3 pyramid, (see ellipsoid 424 in FIG. 4.5 after
examining FIG. 4.5). This is readily noticeable to a student by
removing all but the above-referenced ellipsoids in the 3.times.3
pyramidal configuration. The commonality of latticework structure
is thus demonstrated by adding ellipsoids to a tetrahedral
configuration and then removing ellipsoids to derive a pyramidal
(one-half octahedral) configuration.
FIG. 6.0 shows the cube of FIG. 4.9 from a different angle. Looking
down onto the square orientation indicia of the cube in FIG. 4.9
and 6.0, the 3.times.3 pyramid base is observed while looking onto
the triangle orientation indicia highlights the tetrahedral
configuration, and the common axis orientation circle indicia is
always pointing to the right when either the triangle orientation
indicia or the square orientation indicia are considered to be in
the up position.
Turning now to FIG. 5.1, ellipsoids of the invention are gravity
stacked initially to form a four-sided base layer resting on the
tray 301. The square orientation indicia of each ellipsoid faces up
away from the tray 301. All common axis orientation circle indicia
face the same direction in a recognizable pattern. Successive
layers of ellipsoids are gravity stacked building up from the base
layer. A five layer pyramid configuration 600 of ellipsoids is
formed, the square orientation indicia of each ellipsoid facing
upward, the triangle orientation indicia facing uniformly in one
direction, and the common axis orientation circle indicia uniformly
pointing to the right --as is shown in FIG. 5.6. The pyramid
configuration 600 of FIG. 5.6 may be considered to be one-half of
an octahedral configuration. To complete the octahedron, layers of
ellipsoids may be placed below the base layer. That is, an
arrangement of optionally connected ellipsoids below the base layer
duplicates the arrangement above the base layer --thereby forming
an octahedral configuration.
By comparing the face defined by ellipsoids 401, 402, 405, 406,
407, 403, 404, 408, 409, 410, 411, 412, 413, 414 and 415 of FIG.
5.6 with the back face of FIG. 4.8 which also is the base layer of
FIG. 4.7 before it was rotated forward and also by checking the
ellipsoids stacked in the base layer of 4.7 in FIGS. 4.3, 4.6 and
4.7, it is demonstrated to a student that a tetrahedron face can
lie coextensive with an octahedron face. More specifically in FIG.
5.6, ellipsoids 401, 403, 410, 413 and 415 form a first edge;
ellipsoids 415, 414, 412, 409 and 407 form a second edge; and
ellipsoids 401, 402, 405, 406 and 407 form a third edge along both
the back face of the configuration of FIG. 4.8 and the
above-defined face of FIG. 5.6.
The congruency of faces also demonstrates that tetrahedrons and
octahedrons can be interfit, or merged, to form a common lattice
structure. The congruency of faces similarly demonstrates that
ellipsoids arranged based on a tetrahedral configuration are, in
actuality, arranged the same in relative positioning as ellipsoids
stacked in a latticework founded on a pyramid configuration having
a four-sided base in addition to four faces --the only difference
being one of bearing (or orientation) and not lattice
structure.
Alternatively, these aspects of latticework are demonstrated with
reference again to FIG. 6.0. It is first noted that a student
starts with the tetrahedral configuration having three edges
defined by the ellipsoids 401, 421, 906, 912 and 915; 415, 905,
911, 914 and 915; and 407, 901, 908, 913 and 915 respectively. That
is, the student starts with the arrangement of FIG. 4.8. It is next
noted that laying coextensive against each face of the tetrahedral
configuration is an 1/8th octahedron section readily derivable from
the pyramid (one-half octahedron) configuration 600 of FIG. 5.6.
The 1/8th octahedron sections are derived by cutting the pyramid
configuration 600 with two imaginary planes that are perpendicular
to the surface 303 of tray 301 and that lie along the two diagonals
that are extensions of the cross-section A--A lines. For purposes
of explanation, one 1/8th octahedron section will be examined as
indicated by the cross-section A--A in FIG. 5.6. Ellipsoids 401
through 415 (and ellipsoids positioned thereunder) form a 1/8th
octahedron section. Some of the ellipsoids --such as ellipsoid 415
--are shared by several sections but will nonetheless be maintained
in its integrity with regard to the 401-415 1/8th section.
Examining the tetrahedral configuration of FIG. 4.8 shows that the
upper back face thereof includes ellipsoids 401 through 415 as they
are provided in FIG. 5.6. Further, these fifteen ellipsoidal
elements all have their three orientation marks oriented in the
same directions. Treating the 401-415 ellipsoids on the tetrahedron
face as coexistent with the octahedron face and adding the
ellipsoids to the octahedron face to complete the 1/8th octahedron
section, a corner of the cube in FIGS. 4.9 and 6.0 is formed. The
rear bottom corner of FIG. 6.0 is comprised of the 401-415
ellipsoids and ellipsoids 501, 601, 602, 603, 505, 506 and 514,
(see FIGS. 5.3 and 5.4). In FIGS. 4.9 and 6.0, four correctly
oriented 1/8th octahedron sections are added to the basic
tetrahedral configuration to form the cube. The student will notice
that the 1/8th octahedron section that can be added to a face of
the tetrahedron and form a correctly oriented corner is very
specific. Only one specific correctly oriented 1/8th octahedron
section can be added to any specific face of the two corresponding
tetrahedrons as shown in FIG. 7.0. That is, when an octahedron has
been divided into eight, 1/8th octahedron sections, by passing
planes through the extended cross-section lines of Section A--A of
FIG. 5.6 and the plane of the paper, the student has eight
distinctly differently oriented, 1/8th octahedron sections, each
one of which can be correctly oriented and matched with one of the
eight faces of two matching corresponding tetrahedrons. This also
means that the two matching corresponding tetrahedrons are
distinctly different from each other, one being the `up`
tetrahedron and the other being the `down` tetrahedron,
notwithstanding the initial intuitive feeling that they are the
same when first glancing at FIG. 7.0 and looking at the blocks
themselves. As noted previously, the cube demonstrates the
continuity of latticework when ellipsoids in a tetrahedral
configuration are interfit with ellipsoids in an octahedral
configuration --i.e. that both embody the same latticework
structure.
It will be noted also that the student may start with similarly
dimensioned, similarly oriented ellipsoidal elements arranged in
either one of the two basic configurations, either the basic
tetrahedral configuration (FIG. 4.7) or the basic pyramid
configuration (FIG. 5.6) to demonstrate commonality of latticework
therebetween. In doing so, the student expands the initial
latticework by adding ellipsoidal elements thereto. The student
then selectively removes ellipsoidal elements from the expanded
latticework to form the other basic configuration. This requires
that the student add sufficient ellipsoidal elements to enable the
forming of the other basic configuration. The student may be
assisted by viewing the triangle, square, and circle orientation
indicia applied to the ellipsoidal elements, as suggested in the
FIG. 4.9 and FIG. 7.0 discussion above.
Still a further way of demonstrating the commonality of latticework
between the tetrahedral configuration and the pyramid (one-half
octahedron) configuration relates to FIG. 6.1. FIG. 6.1 depicts a
cuboctahedron 700 formed of thirteen ellipsoids A through M. The
commonality of latticework is readily shown by (a) adding
ellipsoids to the cuboctahedron 700 to form a basic tetrahedral
configuration and (b) also adding ellipsoids to the cuboctahedron
700 to arrive at a basic pyramid configuration --demonstrating that
both have the same nucleus latticework. Alternatively, ellipsoids
are removed from the five-layer configuration of FIG. 4.7 and of
FIG. 5.6 to obtain the cuboctahedron in each case --again
demonstrating common latticework. The positioning of the
cuboctahedron ellipsoids A through M in the two configurations can
be determined by starting with the cuboctahedron 700 in the
rectangular or pyramidal (one-half octahedron) configuration and
following the exact location of ellipsoidal elements denoted by
letters A through M.
The bottom layer in FIG. 6.1 contains ellipsoids A, B, C and D in a
form of a 4-sided square face. In FIG. 5.2 the student sees that
ellipsoids 503, 504, 501 and 502 are equal to A, B, C and D of the
cuboctahedron 700, and they are in the correct orientation. In FIG.
4.3 the student sees that ellipsoids 404, 408, 422 and 900 are
equal to A, B, C and D of the cuboctahedron 700, and they are in
the correct orientation.
The second layer of FIG. 6.1 contains ellipsoids E, F, G, H and I
in the form of a cross. In FIG. 5.3 the student sees that
ellipsoids 512, 508, 509, 510 and 506 are equal to E, F, G, H and I
of the cuboctahedron 700, and they are in the correct orientation.
In FIG. 4.5 the student sees that ellipsoids 411, 423, 424, 902 and
907 are equal to E, F, G, H and I of cuboctahedron 700, and they
are in the correct orientation.
The third and top layer of FIG. 6.1 contains ellipsoids J, K, L and
M in the form of a square face. In FIG. 5.4 the student sees that
ellipsoids 516, 517, 514 and 515 are equal to J, K, L and M of the
cuboctahedron 700, and they are in the correct orientation. In FIG.
4.6 the student sees that ellipsoids 903, 904, 909 and 910 are
equal to J, K, L and M of cuboctahedron 700, and they are in the
correct orientation.
The cuboctahedron 700 in FIGS. 5.2, 5.3, 5.4 and 5.5 is also of
significance in demonstrating that the common latticework is
arranged in imaginary thirteen nonparallel planes. The student can
examine FIGS. 5.2, 5.3, 5.4 and 5.5 and identify the imaginary
thirteen nonparallel planes that pass through the center ellipsoid
509 as follows:
(1) that plane that passes through the centerpoints of ellipsoids
509, 505, 506, 507, 508, 510, 511, 512 and 513 --this is surface
303 of tray 301 when the cuboctahedron 700 is in the rectangular
configuration and the (x,y) plane in three-dimensional
coordinates;
(2) that plane that passes through the centerpoints of ellipsoids
509, 508, 510, 518 and 519 --this is the (y,z) plane in
three-dimensional coordinates;
(3) that plane that passes through the centerpoints of ellipsoids
509, 506, 512, 519 and 414 --this is the (x,z) plane in
three-dimensional coordinates;
(4) that plane that passes through the centerpoints of ellipsoids
509, 508, 510, 501, 502, 516 and 517 --this plane is parallel to
surface 303 when in the triangular or tetrahedral
configuration;
(5) that plane that passes through the centerpoints of ellipsoids
509, 508, 510, 503, 504, 514 and 515;
(6) that plane that passes through the centerpoints of ellipsoids
509, 506, 512, 502, 504, 514 and 516;
(7) that plane that passes through the centerpoints of ellipsoids
509, 506, 512, 501, 503, 515 and 517;
(8) that plane that passes through the centerpoints of ellipsoids
509, 505, 513, 501, 504, 514 and 517;
(9) that plane that passes through the centerpoints of ellipsoids
509, 505, 513, 503 and 515;
(10) that plane that passes through the centerpoints of ellipsoids
509, 505, 513 502 and 516;
(11) that plane that passes through the centerpoints of ellipsoids
509, 507, 511, 502, 503, 515 and 516;
(12) that plane that passes through the centerpoints of ellipsoids
509, 507, 511, 501 and 517;
(13) that plane that passes through the centerpoints of ellipsoids
509, 507, 511, 504 and 514.
These imaginary thirteen nonparallel planes define the general
common latticework structure that results when ellipsoids of
influence are closely packed under the aforementioned four
conditions.
FIG. 7.0 shows a corresponding octahedron being merged with a pair
of matching corresponding tetrahedrons without any ellipsoidal
elements being shown. FIG. 7.0, that is, shows a plurality of
spacepoints 701, 702, 703 and 721, representing a first `up`
corresponding tetrahedron; and 722, 723, 724 and 704, representing
the `down` or inverted matching corresponding tetrahedron; and 821,
822, 823, 802, 803 and 804, representing their matching
corresponding octahedron, which spacepoints define merged
interfitting elements. The spacepoints define a latticework
structure, such as a crystal lattice or the like. Preferably, the
spacepoints correspond to the centerpoints of ellipsoidal elements
--such as the ellipsoids illustrated in FIGS. 4.0 through 6.1.
Also, preferably, all ellipsoidal elements are similarly
dimensioned and similarly oriented as suggested in the ellipsoidal
embodiment above.
However other structural members, such as the corresponding
tetrahedrons and corresponding octahedrons with unique
corner-to-corner distance ratios equal to center-to-center
ellipsoid ratios as set forth in Table II, may be employed to
define the center-to-center distances of ellipsoidal elements when
imaginary thirteen nonparallel planes are involved. When twinning
of any of the imaginary thirteen nonparallel planes is involved
then the corresponding tetrahedrons and the corresponding
octahedrons are the preferable embodiment of the invention. This
feature is better understood by examining the unique ratios of
center-to-center distances of ellipsoids in Table I and the
corresponding unique corner-to-corner distances of tetrahedrons and
octahedrons in Table II.
In Table I Sections (a) through (d), a variety of types of
ellipsoid sets are listed together with the center-to-center
distances that correspond to the sets. By examining FIGS. 4.0
through 6.1, it will be recognized that the center-to-center
distances between adjacent touching ellipsoids is related to the
lengths of the three orthogonal axes of symmetry of the similar
ellipsoidal elements when the common axis and the location of
either orientation mark are known. In all of the unique sets of
ellipsoids referred to in Table I Sections (a) through (d) the
length of the first axis of the corresponding ellipsoid is
arbitrarily given the unit dimension `D`. This first axis is also
the common axis. The length of the second axis is given directly in
Table I Sections (a) through (d) after the student studies the
general arrangement required by the remaining center-to-center
distances given in ratios of the unit distance `D`. The orientation
of the third axis is also determined by said general arrangement.
The student may then vary the length of the third axis to make the
remaining three or four equal length center-to-center distances
match that distance as set forth in Table I Sections (a) through
(d). Conversely, if a desired latticework is sought, the length of
the orthogonal axes of symmetry may be selected accordingly. FIGS.
4.0 through 6.1 illustrate the equilateral ellipsoid set.
In Table II Sections (a) through (d), the distances between
spacepoints in FIG. 7.0 are associated with pairs of matching
corresponding tetrahedrons and their matching corresponding
octahedron sets. While the spacepoint distances are preferably
altered by employing ellipsoidal elements of selected dimensions,
it is also contemplated that each edge shown in FIG. 7.0 be an edge
on a set of matching corresponding tetrahedrons and their matching
corresponding octahedron.
For example, to achieve the set of snowflake blocks, according to
Table II Section (b), the tetrahedron and octahedron snowflake
blocks are made with just one congruent triangular face. Each
tetrahedron snowflake block has four of these congruent faces and
the octahedron snowflake block has eight of these congruent faces.
The snowflake congruent face has one edge that has arbitrarily been
given a unit `D` length between the spacepoint pairs as indicated
in Table II Section (b). The other two edges of the snowflake
congruent triangular face are of equal length and Table II Section
(b) shows the spacepoint-to-spacepoint distance as a ratio of the
given unit distance `D` length. The snowflake congruent face has
two edge lengths that are equal to the ratio of the square root of
5/4ths, substantially equal to (1.11803) multiplied by the unit
distance `D` length of the third edge. By using ratios to dimension
the exact center-to-center distance between ellipsoids in Table I
and exact spacepoint-to-spacepoint distance between corners in
Table II the ellipsoids and the blocks can be made any size that is
suitable to implement the invention. The ellipsoid numbers used in
Table I and spacepoints used in Table II, correspond to the numbers
used in FIG. 4.2 and FIG. 7.0 respectively.
The center-to-center distance ratios in Table I, and
spacepoint-to-spacepoint distance ratios in Table II are unique and
can be used to demonstrate simultaneous twinning in more than one
of the imaginary thirteen nonparallel planes.
The ellipsoidal element embodiment, it is noted is more convenient,
more demonstrative, and preferable in showing not only distances
but also gravity stacking.
Any of various latticeworks --of tetrahedral and octahedral
configurations --can be formed with ellipsoidal elements or with
the corresponding tetrahedron and octahedron blocks as structure
members for defining the spacing between spacepoints, especially
according to the sets listed in Table II Sections (a) through (d).
Moreover, the orientation of the imaginary thirteen nonparallel
planes may vary but the planes still remain the exclusive set of
imaginary thirteen nonparallel planes as long as no twinning
occurs.
It is a further object of the invention, when using the tetrahedron
and octahedron blocks of Table II, to demonstrate that the two
touching congruent faces may be used to determine if twinning of
the latticework structure is occurring. If one of the two touching
congruent faces is on a tetrahedron block and the other touching
congruent face is on an octahedron block, then no twinning of the
latticework is occurring on that set of congruent faces.
Conversely, if both touching congruent faces are on either two
tetrahedron blocks or two octahedron blocks then twinning is
occurring on that congruent face, excepting only when the set of
blocks are of such dimensions or ratios that the octahedron block
can be constructed from four tetrahedron blocks --such as is the
case with the snowflake "SF4" blocks --in which case twinning may
or may not be occurring since it is possible to connect four "SF4"
tetrahedrons and cause these four tetrahedrons to appear to be a
corresponding matching octahedron.
It is also an object of the invention to demonstrate that using one
set of corresponding ellipsoids, which define one unique set of
imaginary thirteen nonparallel planes in a space latticework, when
closely packed under the aforementioned four conditions, which
further define one unique set of one matching corresponding `up`
tetrahedron and one matching corresponding `down` tetrahedron and
their matching octahedron, when this set of ellipsoids can be
simultaneously twinned in more than one of the imaginary thirteen
nonparallel planes --such as with the snowflake blocks --it is
possible to create literally millions of combinations of `domains`
of imaginary thirteen nonparallel plane space latticework where the
combination of `domains` make the resulting visible crystal
structure appear completely different than just one simple
imaginary thirteen nonparallel plane latticework structure made
from just one ellipsoid of influence which has been twinned into
many `domains`.
Other improvements, modifications, and embodiments will become
apparent to one of ordinary skill in the art upon review of this
disclosure. Such improvements, modifications and embodiments are
considered to be within the scope of this invention as defined in
the following claims. For example, although it is preferred that
the ellipsoidal elements be geometric ellipsoids, it is
contemplated that the elements may be constructed with complex
combinations of ellipsoidal surfaces that have been merged together
and further that have a common axis that is not on one of the
orthogonal axes of symmetry of the ellipsoidal elements. The
coupling devices between the faces of the tetrahedrons and the
octahedrons can be velcro, magnetic, pressure sensitive material,
or any other suitable device or means of connecting the two
congruent faces.
Another example would be to have the ellipsoid set be stacked by
gravity as set forth in the invention and then expand the
ellipsoidal surfaces into the interstilial spaces equally until the
surfaces met the corresponding expanding surfaces of the adjacent
ellipsoid. This creates a corresponding set of blocks with flat
surfaces that can be stacked the way the ellipsoids of influence
are stacked.
Further, the tetrahedron and octahedron blocks of the block sets as
set forth in Table II Sections (a) through (d) can be divided in
such a manner that a plane is made equidistant from each corner
spacepoint and thus each tetrahedron block is cut into four pieces
and each octahedron block is cut into six pieces that demonstrate
the same planes above described for the corresponding ellipsoid.
This again demonstrates the logical field of influence surrounding
the unique ellipsoid sets of Table I and unique tetrahedron and
octahedron block sets of Table II.
Also the tetrahedron and octahedron blocks of the block sets may be
divided into parts along one of the thirteen nonparallel planes
demonstrated by the invention, starting at one of the corner
spacepoints and proceeding equidistant from the nearest remaining
corner spacepoints, to demonstrate to the student how an ellipsoid
of influence and the corresponding tetrahedrons and octahedrons
could logically describe customary crystal latticework structures.
Other similar solid shapes may also be employed in accordance with
the claimed invention.
Conversely, a plurality of corresponding dimensioned blocks
consisting of merged combinations of two or more corresponding
tetrahedrons and octahedrons where no twinning is occurring are
considered to be in the subject matter of the invention; for one
specific example,
an additional plurality of corresponding dimensioned blocks
consisting of a merged first tetrahedron and octahedron along
congruent faces containing spacepoints 721, 703 and 702 and 821,
803 and 802 and a second tetrahedron with the said octahedron along
congruent faces containing spacepoints 721, 702 and 701 and 823,
804 and 803, where said spacepoints are as numbered in FIG.
7.0.
Further still, a plurality of corresponding dimensioned blocks
consisting of merged combinations of two or more corresponding
tetrahedrons and/or octahedrons where twinning is occurring are
considered to be in the subject matter of the invention;
for example, an additional plurality of corresponding dimensioned
blocks consisting of six merged `Snowflake` tetrahedrons where six
common axis edges 701-702 are merged and centered with a vertical
bearing with the six opposite edges 703-721 away from said centered
merged edges 701-702 in the shape of a hexagon when viewed along
the 701-702 center axis --thus starting six `domains` of imaginary
13 plane latticework with twinning in all six planes where six
unmerged tetrahedrons would normally be touching;
another example is where three `Snowflake` octahedrons are merged
such that three edges 821-803 are merged and centered with a
vertical bearing with the edges 823-822-802 in the shape of a
hexagon when viewed along the 821-803 center axis --thus starting
three `domains` of imaginary 13 plane latticework with twinning in
all three planes where three unmerged octahedrons would normally be
touching;
still another example is where 20 `Icosahedron` tetrahedrons are
merged such that twenty corners numbered 721 or 704 are touching
forming a seed icosahedron --thus starting 20 `domains` of 13 plane
latticework with twinning in all 30 planes where 20 unmerged
tetrahedrons would normally be touching;
further still is where 5 `Icosahedron` tetrahedrons are merged such
that 5 corners numbered 721 or 704 are touching and 5 edges like
721-703 or 704-722 are also merged in a central axis --thus making
a `cap like` combination that fits over the 5 twinned octahedrons
that rest on each of the 12 points of the icosahedron.
TABLE I ______________________________________ Ellipsoids such that
in FIG. 4.2 the Center-to-Center Distance Between Ellipsoid Numbers
______________________________________ Section (a) 401 and 402 403
and 401; 403 and 402 Ellipsoid 421 and 403 421 and 401; 421 and 402
Set are and are equal to ______________________________________
Equilateral Equal to Distance `D` Ellipsoids Distance `D`
______________________________________ Section (b) 401 and 402 403
and 401; 403 and 402 Ellipsoid 421 and 403 421 and 401; 421 and 402
Set are and are equal to ______________________________________
Snowflake Equal to 1.11803 times Ellipsoids Distance `D` Distance
`D` "SF3" Equal to 0.76376 times Ellipsoids Distance `D` Distance
`D` "SF4" Equal to 0.86603 times Ellipsoids Distance `D` Distance
`D` "SF5" Equal to 0.98672 times Ellipsoids Distance `D` Distance
`D` "SF7" Equal to 1.25618 times Ellipsoids Distance `D` Distance
`D` "SF8" Equal to 1.39897 times Ellipsoids Distance `D` Distance
`D` "SF9" Equal to 1.54504 times Ellipsoids Distance `D` Distance
`D` "SF10" Equal to 1.69353 times Ellipsoids Distance `D` Distance
`D` "SF11" Equal to 1.84382 times Ellipsoids Distance `D` Distance
`D` "SF12" Equal to 1.99551 times Ellipsoids Distance `D` Distance
`D` ______________________________________ Section (c) 401 and 402
401 and 421 402 and 403 402 and 421 Ellipsoid 403 and 401 403 and
421 Set are and are equal to ______________________________________
Cube Equal to 0.70711 times Ellipsoids Distance `D` Distance `D`
Icosahedron Equal to 0.95106 times Ellipsoids Distance `D` Distance
`D` Diamond Equal to 0.61237 times Ellipsoids Distance `D` Distance
`D` ______________________________________ Section (d) 401 and 403
and 401; 403 and 402 Ellipsoid 402 421 and 403 421 and 401; 421 and
402 Set are are equal to are equal to
______________________________________ "SF3 .times. 4" Equal to
0.57735 times 0.64550 times Ellipsoids Distance Distance `D`
Distance `D` `D` "SF3 .times. 5" Equal to 0.41947 times 0.61427
times Ellipsoids Distance Distance `D` Distance `D` `D` "SF3
.times. 6" Equal to 0.33333 times 0.60093 times Ellipsoids Distance
Distance `D` Distance `D` `D` "SF3 .times. 7" Equal to 0.27804
times 0.59385 times Ellipsoids Distance Distance `D` Distance `D`
`D` "SF3 .times. 8" Equal to 0.23915 times 0.58960 times Ellipsoids
Distance Distance `D` Distance `D` `D` "SF3 .times. 9" Equal to
0.21014 times 0.58683 times Ellipsoids Distance Distance `D`
Distance `D` `D` "SF3 .times. 10" Equal to 0.18759 times 0.58492
times Ellipsoids Distance Distance `D` Distance `D` `D` "SF3
.times. 11" Equal to 0.16953 times 0.58354 times Ellipsoids
Distance Distance `D` Distance `D` `D` "SF3 .times. 12" Equal to
0.15470 times 0.58251 times Ellipsoids Distance Distance `D`
Distance `D` `D` "SF4 .times. 5" Equal to 0.72654 times 0.79496
times Ellipsoids Distance Distance `D` Distance `D` `D` "SF4
.times. 6" Equal to 0.57735 times 0.76376 times Ellipsoids Distance
Distance `D` Distance `D` `D` "SF4 .times. 7" Equal to 0.48157
times 0.74698 times Ellipsoids Distance Distance `D` Distance `D`
`D` "SF4 .times. 8" Equal to 0.41421 times 0.73681 times Ellipsoids
Distance Distance `D` Distance `D` `D` "SF4 .times. 9" Equal to
0.36397 times 0.73015 times Ellipsoids Distance Distance `D`
Distance `D` `D` "SF4 .times. 10" Equal to 0.32492 times 0.72553
times Ellipsoids Distance Distance `D` Distance `D` `D` "SF4
.times. 11" Equal to 0.29363 times 0.72219 times Ellipsoids
Distance Distance `D` Distance `D` `D` "SF4 .times. 12" Equal to
0.26795 times 0.71969 times Ellipsoids Distance Distance `D`
Distance `D` `D` "SF5 .times. 6" Equal to 0.79465 times 0.93887
times Ellipsoids Distance Distance `D` Distance `D` `D` "SF5
.times. 7" Equal to 0.66283 times 0.91293 times Ellipsoids Distance
Distance `D` Distance `D` `D` "SF5 .times. 8" Equal to 0.57012
times 0.89714 times Ellipsoids Distance Distance `D` Distance `D`
`D` "SF5 .times. 9" Equal to 0.50096 times 0.88676 times Ellipsoids
Distance Distance `D` Distance `D` `D` "SF5 .times. 10" Equal to
0.44721 times 0.87955 times Ellipsoids Distance Distance `D`
Distance `D` `D` "SF5 .times. 11" Equal to 0.40414 times 0.87432
times Ellipsoids Distance Distance `D` Distance `D` `D` "SF5
.times. 12" Equal to 0.36880 times 0.87041 times Ellipsoids
Distance Distance `D` Distance `D` `D` "SF6 .times. 7" Equal to
0.83411 times 1.08348 times Ellipsoids Distance Distance `D`
Distance `D` `D` "SF6 .times. 8"
Equal to 0.71744 times 1.06239 times Ellipsoids Distance Distance
`D` Distance `D` `D` "SF6 .times. 9" Equal to 0.63041 times 1.04850
times Ellipsoids Distance Distance `D` Distance `D` `D` "SF6
.times. 10" Equal to 0.56278 times 1.03884 times Ellipsoids
Distance Distance `D` Distance `D` `D` "SF6 .times. 11" Equal to
0.50858 times 1.03182 times Ellipsoids Distance Distance `D`
Distance `D` `D` "SF6 .times. 12" Equal to 0.46410 times 1.02657
times Ellipsoids Distance Distance `D` Distance `D` `D` "SF7
.times. 8" Equal to 0.86012 times 1.23002 times Ellipsoids Distance
Distance `D` Distance `D` `D` "SF7 .times. 9" Equal to 0.75579
times 1.21276 times Ellipsoids Distance Distance `D` Distance `D`
`D` "SF7 .times. 10" Equal to 0.67470 times 1.20075 times
Ellipsoids Distance Distance `D` Distance `D` `D` "SF7 .times. 11"
Equal to 0.60972 times 1.19203 times Ellipsoids Distance Distance
`D` Distance `D` `D` "SF7 .times. 12" Equal to 0.55640 times
1.18549 times Ellipsoids Distance Distance `D` Distance `D` `D`
"SF8 .times. 9" Equal to 0.87870 times 1.37845 times Ellipsoids
Distance Distance `D` Distance `D` `D` "SF8 .times. 10" Equal to
0.78443 times 1.36416 times Ellipsoids Distance Distance ` D`
Distance `D` `D` "SF8 .times. 11" Equal to 0.70888 times 1.35378
times Ellipsoids Distance Distance `D` Distance `D` `D` "SF8
.times. 12" Equal to 0.64689 times 1.34600 times Ellipsoids
Distance Distance `D` Distance `D` `D` "SF9 .times. 10" Equal to
0.89271 times 1.52853 times Ellipsoids Distance Distance `D`
Distance `D` `D` "SF9 .times. 11" Equal to 0.80673 times 1.51653
times Ellipsoids Distance Distance `D` Distance `D` `D` "SF9
.times. 12" Equal to 0.73618 times 1.50753 times Ellipsoids
Distance Distance `D` Distance `D` `D` "SF10 .times. 11" Equal to
0.90369 times 1.67994 times Ellipsoids Distance Distance `D`
Distance `D` `D` "SF10 .times. 12" Equal to 0.82466 times 1.66975
times Ellipsoids Distance Distance `D` Distance `D` `D` "SF11
.times. 12" Equal to 0.91255 times 1.83245 times Ellipsoids
Distance Distance `D` Distance `D` `D`
______________________________________
TABLE II ______________________________________ Tetrahedrons and
Octahedrons such that in FIG. 7.0 the Spacepoint-to-Spacepoint
Distance Between Corners Numbered
______________________________________ Section (a) 701 and 702 703
and 701; 703 and 702 703 and 721 721 and 701; 721 and 702 723 and
724 722 and 723; 722 and 724 704 and 722 704 and 723; 704 and 724
803 and 821 823 and 803; 823 and 804 Tetrahedron 804 and 822 823
and 821; 823 and 822 and 803 and 804 802 and 803; 802 and 804
Octahedron 821 and 822 802 and 821; 802 and 822 Sets are and are
equal to ______________________________________ Equilateral Equal
to Distance `D` Blocks Distance `D`
______________________________________ Section (b) 701 and 702 703
and 701; 703 and 702 703 and 721 721 and 701; 721 and 702 723 and
724 722 and 723; 722 and 724 704 and 722 704 and 723; 704 and 724
803 and 821 823 and 803; 823 and 804 Tetrahedron 804 and 822 823
and 821; 823 and 822 and 803 and 804 802 and 803; 802 and 804
Octahedron 821 and 822 802 and 821; 802 and 822 Sets are and are
equal to ______________________________________ Snowflake Equal to
1.11803 times Blocks Distance `D` Distance `D` "SF3" Equal to
0.76376 times Blocks Distance `D` Distance `D` "SF4" Equal to
0.86603 times Blocks Distance `D` Distance `D` "SF5" Equal to
0.98672 times Blocks Distance `D` Distance `D` "SF7" Equal to
1.25618 times Blocks Distance `D` Distance `D` "SF8" Equal to
1.39897 times Blocks Distance `D` Distance `D` "SF9" Equal to
1.54504 times Blocks Distance `D` Distance `D` "SF10" Equal to
1.69353 times Blocks Distance `D` Distance `D` "SF11" Equal to
1.84382 times Blocks Distance ` D` Distance `D` "SF12" Equal to
1.99551 times Blocks Distance `D` Distance `D`
______________________________________ Section (c) 701 and 702; 702
and 703 721 and 701; 721 and 702 703 and 701; 723 and 724 721 and
703; 704 and 722 724 and 722; 722 and 723 704 and 723; 704 and 724
Tetrahedron 821 and 822; 822 and 823 802 and 821; 802 and 822 and
823 and 821; 802 and 803 803 and 821; 803 and 823 Octahedron 803
and 804; 804 and 803 804 and 822; 804 and 823 Sets are are equal to
______________________________________ Cube Equal to 0.70711 times
Blocks Distance `D` Distance `D` Icosahedron Equal to 0.95106 times
Blocks Distance `D` Distance `D` Diamond Equal to 0.61237 times
Blocks Distance `D` Distance `D`
______________________________________ Section (d) 701 and 721 and
701; 721 and 702 702 703 and 701; 703 and 702 723 and 722 and 723;
722 and 724 724 704 and 723; 704 and 724 821 and 721 and 703 802
and 821; 802 and 822 Tetrahedron 822 704 and 722 802 and 803; 802
and 804 and 803 and 803 and 821 823 and 821; 823 and 822 Octahedron
804 804 and 822 823 and 803; 823 and 804 Sets are are equal to are
equal to ______________________________________ "SF3 .times. 4"
Equal to 0.57735 times 0.64550 times Blocks Distance Distance `D`
Distance `D` `D` "SF3 .times. 5" Equal to 0.41947 times 0.61427
times Blocks Distance Distance `D` Distance `D` `D` "SF3 .times. 6"
Equal to 0.33333 times 0.60093 times Blocks Distance Distance `D`
Distance `D` `D` "SF3 .times. 7" Equal to 0.27804 times 0.59385
times Blocks Distance Distance `D` Distance `D` `D` "SF3 .times. 8"
Equal to 0.23915 times 0.58960 times Blocks Distance Distance `D`
Distance `D` ` D` "SF3 .times. 9" Equal to 0.21014 times 0.58683
times Blocks Distance Distance `D` Distance `D` `D` "SF3 .times.
10" Equal to 0.18759 times 0.58492 times Blocks Distance Distance
`D` Distance `D` `D` "SF3 .times. 11" Equal to 0.16953 times
0.58354 times Blocks Distance Distance `D` Distance `D` `D` "SF3
.times. 12" Equal to 0.15470 times 0.58251 times Blocks Distance
Distance `D` Distance `D` `D` "SF4 .times. 5" Equal to 0.72654
times 0.79496 times Blocks Distance Distance `D` Distance `D` `D`
"SF4 .times. 6" Equal to 0.57735 times 0.76376 times Blocks
Distance Distance `D` Distance `D` `D` "SF4 .times. 7" Equal to
0.48157 times 0.74698 times Blocks Distance Distance `D` Distance
`D` `D` "SF4 .times. 8" Equal to 0.41421 times 0.73681 times Blocks
Distance Distance `D` Distance `D` `D` "SF4 .times. 9" Equal to
0.36397 times 0.73015 times Blocks Distance Distance `D` Distance
`D` `D` "SF4 .times. 10" Equal to 0.32492 times 0.72553 times
Blocks Distance Distance `D` Distance `D` `D` "SF4 .times. 11"
Equal to 0.29363 times 0.72219 times Blocks Distance Distance `D`
Distance `D` `D` "SF4 .times. 12" Equal to 0.26795 times 0.71969
times Blocks Distance Distance `D` Distance `D` `D` "SF5 .times. 6"
Equal to 0.79465 times 0.93887 times Blocks Distance Distance `D`
Distance `D` `D` "SF5 .times. 7" Equal to 0.66283 times 0.91293
times Blocks Distance Distance `D` Distance `D` `D` "SF5 .times. 8"
Equal to 0.57012 times 0.89714 times Blocks Distance Distance `D`
Distance ` D` `D` "SF5 .times. 9" Equal to 0.50096 times 0.88676
times Blocks Distance Distance `D` Distance `D` `D` "SF5 .times.
10" Equal to 0.44721 times 0.87955 times Blocks Distance Distance
`D` Distance `D` `D` "SF5 .times. 11" Equal to 0.40414 times
0.87432 times Blocks Distance Distance `D` Distance `D` `D` "SF5
.times. 12" Equal to 0.36880 times 0.87041 times Blocks Distance
Distance `D` Distance `D` `D` "SF6 .times. 7" Equal to 0.83411
times 1.08348 times Blocks Distance Distance `D` Distance `D` `D`
"SF6 .times. 8"
Equal to 0.71744 times 1.06239 times Blocks Distance Distance `D`
Distance `D` `D` "SF6 .times. 9" Equal to 0.63041 times 1.04850
times Blocks Distance Distance `D` Distance `D` `D` "SF6 .times.
10" Equal to 0.56278 times 1.03884 times Blocks Distance Distance
`D` Distance `D` `D` "SF6 .times. 11" Equal to 0.50858 times
1.03182 times Blocks Distance Distance `D` Distance `D` `D` "SF6
.times. 12" Equal to 0.46410 times 1.02657 times Blocks Distance
Distance `D` Distance `D` `D` "SF7 .times. 8" Equal to 0.86012
times 1.23002 times Blocks Distance Distance `D` Distance `D` `D`
"SF7 .times. 9" Equal to 0.75579 times 1.21276 times Blocks
Distance Distance `D` Distance `D` `D` "SF7 .times. 10" Equal to
0.67470 times 1.20075 times Blocks Distance Distance `D` Distance
`D` `D` "SF7 .times. 11" Equal to 0.60972 times 1.19203 times
Blocks Distance Distance `D` Distance `D` `D` "SF7 .times. 12"
Equal to 0.55640 times 1.18549 times Blocks Distance Distance `D`
Distance `D` `D` "SF8 .times. 9" Equal to 0.87870 times 1.37845
times Blocks Distance Distance `D` Distance `D` `D` "SF8 .times.
10" Equal to 0.78443 times 1.36416 times Blocks Distance Distance
`D` Distance `D` `D` "SF8 .times. 11" Equal to 0.70888 times
1.35378 times Blocks Distance Distance `D` Distance `D` `D` "SF8
.times. 12" Equal to 0.64689 times 1.34600 times Blocks Distance
Distance `D` Distance `D` `D` "SF9 .times. 10" Equal to 0.89271
times 1.52853 times Blocks Distance Distance `D` Distance `D` `D`
"SF9 .times. 11" Equal to 0.80673 times 1.51653 times Blocks
Distance Distance `D` Distance `D` `D` "SF9 .times. 12" Equal to
0.73618 times 1.50753 times Blocks Distance Distance `D` Distance
`D` `D` "SF10 .times. 11" Equal to 0.90369 times 1.67994 times
Blocks Distance Distance `D` Distance `D` `D` "SF10 .times. 12"
Equal to 0.82466 times 1.66975 times Blocks Distance Distance `D`
Distance `D` `D` "SF11 .times. 12" Equal to 0.91255 times 1.83245
times Blocks Distance Distance `D` Distance `D` `D`
______________________________________
* * * * *